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Mathematics LibreTexts

11-4. Series and Their Notations

Series and Their Notations
  • Use summation notation.
  • Use the formula for the sum of the first n terms of an arithmetic series.
  • Use the formula for the sum of the first n terms of a geometric series.
  • Use the formula for the sum of an infinite geometric series.
  • Solve annuity problems.

A couple decides to start a college fund for their daughter. They plan to invest $50 in the fund each month. The fund pays 6% annual interest, compounded monthly. How much money will they have saved when their daughter is ready to start college in 6 years? In this section, we will learn how to answer this question. To do so, we need to consider the amount of money invested and the amount of interest earned.

Using Summation Notation

To find the total amount of money in the college fund and the sum of the amounts deposited, we need to add the amounts deposited each month and the amounts earned monthly. The sum of the terms of a sequence is called a series. Consider, for example, the following series.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>3</mn><mo>+</mo><mn>7</mn><mo>+</mo><mn>11</mn><mo>+</mo><mn>15</mn><mo>+</mo><mn>19</mn><mo>+</mo><mn>...</mn></mrow></annotation-xml></semantics></math>

The <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mtext>th </mtext></mrow></annotation-xml></semantics></math>partial sum of a series is the sum of a finite number of consecutive terms beginning with the first term. The notation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> S n  represents the partial sum.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr><mtd><msub><mi>S</mi></msub></mtd></mtr></mtable></annotation-xml></semantics></math> 1 =3 S 2 =3+7=10 S 3 =3+7+11=21 S 4 =3+7+11+15=36

Summation notation is used to represent series. Summation notation is often known as sigma notation because it uses the Greek capital letter sigma, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>Σ</mtext><mo>,</mo></mrow></annotation-xml></semantics></math> to represent the sum. Summation notation includes an explicit formula and specifies the first and last terms in the series. An explicit formula for each term of the series is given to the right of the sigma. A variable called the index of summation is written below the sigma. The index of summation is set equal to the lower limit of summation, which is the number used to generate the first term in the series. The number above the sigma, called the upper limit of summation, is the number used to generate the last term in a series.

Explanation of summation notion as described in the text.

If we interpret the given notation, we see that it asks us to find the sum of the terms in the series<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> a k =2k for <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow></annotation-xml></semantics></math> through <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>k</mi><mo>=</mo><mn>5.</mn><mtext> </mtext></mrow></annotation-xml></semantics></math> We can begin by substituting the terms for <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>k</mi></annotation-xml></semantics></math> and listing out the terms of this series.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mtable columnalign="left"><mtr><mtd><mrow/></mtd></mtr><mtr><mtd><msub><mi>a</mi></msub></mtd></mtr></mtable></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 1 =2(1)=2 a 2 =2(2)=4 a 3 =2(3)=6 a 4 =2(4)=8 a 5 =2(5)=10

We can find the sum of the series by adding the terms:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics></math> k=1 5 2k =2+4+6+8+10=30
Summation Notation

The sum of the first<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics></math>terms of a series can be expressed in summation notation as follows:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics></math> k=1 n a k

This notation tells us to find the sum of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics></math> k from <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow></annotation-xml></semantics></math> to <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>k</mi><mo>=</mo><mi>n</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>k</mi><mtext> </mtext></mrow></annotation-xml></semantics></math> is called the index of summation, 1 is the lower limit of summation, and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics></math> is the upper limit of summation.

Does the lower limit of summation have to be 1?

No. The lower limit of summation can be any number, but 1 is frequently used. We will look at examples with lower limits of summation other than 1.

Given summation notation for a series, evaluate the value.

  1. Identify the lower limit of summation.
  2. Identify the upper limit of summation.
  3. Substitute each value of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>k</mi></annotation-xml></semantics></math> from the lower limit to the upper limit into the formula.
  4. Add to find the sum.
Using Summation Notation

Evaluate<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics></math> k=3 7 k 2 .

According to the notation, the lower limit of summation is 3 and the upper limit is 7. So we need to find the sum of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>k</mi></msup></mrow></annotation-xml></semantics></math> 2 from <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>k</mi><mo>=</mo><mn>3</mn></mrow></annotation-xml></semantics></math> to <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>k</mi><mo>=</mo><mn>7.</mn></mrow></annotation-xml></semantics></math> We find the terms of the series by substituting <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>k</mi><mo>=</mo><mn>3</mn><mtext>,</mtext><mn>4</mn><mtext>,</mtext><mn>5</mn><mtext>,</mtext><mn>6</mn><mtext>,</mtext><mtext> </mtext></mrow></annotation-xml></semantics></math> and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>7</mn></mrow></annotation-xml></semantics></math> into the function <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>k</mi></msup></mrow></annotation-xml></semantics></math> 2 . We add the terms to find the sum.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> k=3 7 k 2 = 3 2 + 4 2 + 5 2 + 6 2 + 7 2 =9+16+25+36+49 =135

Evaluate<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics></math> k=2 5 (3k–1) .

38

Using the Formula for Arithmetic Series

Just as we studied special types of sequences, we will look at special types of series. Recall that an arithmetic sequence is a sequence in which the difference between any two consecutive terms is the common difference,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>d</mi><mo>.</mo></mrow></annotation-xml></semantics></math> The sum of the terms of an arithmetic sequence is called an arithmetic series. We can write the sum of the first <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics></math> terms of an arithmetic series as:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> n = a 1 +( a 1 +d)+( a 1 +2d)+...+( a n –d)+ a n .

We can also reverse the order of the terms and write the sum as

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> n = a n +( a n –d)+( a n –2d)+...+( a 1 +d)+ a 1 .

If we add these two expressions for the sum of the first <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi></mrow></annotation-xml></semantics></math>terms of an arithmetic series, we can derive a formula for the sum of the first <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi></mrow></annotation-xml></semantics></math> terms of any arithmetic series.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><msub/></mrow></mtd></mtr></mtable></mrow></mfrac></mrow></annotation-xml></semantics></math> S n = a 1 +( a 1 +d)+( a 1 +2d)+...+( a n –d)+ a n +   S n = a n +( a n –d)+( a n –2d)+...+( a 1 +d)+ a 1 2 S n =( a 1 + a n)+( a 1 + a n )+...+( a 1 + a n )

Because there are <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics></math> terms in the series, we can simplify this sum to

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>2</mn><msub/></mrow></annotation-xml></semantics></math> S n =n( a 1 + a n ).

We divide by 2 to find the formula for the sum of the first <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi></mrow></annotation-xml></semantics></math> terms of an arithmetic series.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> n = n( a 1 + a n ) 2
Formula for the Sum of the First n Terms of an Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence. The formula for the sum of the first <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics></math> terms of an arithmetic sequence is

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> n = n( a 1 + a n ) 2

Given terms of an arithmetic series, find the sum of the first <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi></mrow></annotation-xml></semantics></math> terms.

  1. Identify <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics></math> 1 and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics></math> n .
  2. Determine <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mo>.</mo></mrow></annotation-xml></semantics></math>
  3. Substitute values for <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics></math> 1 ,  a n ,  and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>n</mi><mtext> </mtext></mrow></annotation-xml></semantics></math> into the formula <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> n = n( a 1 + a n ) 2 .
  4. Simplify to find <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> n .
Finding the First n Terms of an Arithmetic Series

Find the sum of each arithmetic series.

  1. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 + 32</mtext></mrow></annotation-xml></semantics></math>
  2. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>20 + 15 + 10 +…+ −50</mtext></mrow></annotation-xml></semantics></math>
  3. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics></math> k=1 12 3k−8
  1. We are given <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics></math> 1 =5 and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> a n =32.

    Count the number of terms in the sequence to find <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mo>=</mo><mn>10.</mn></mrow></annotation-xml></semantics></math>

    Substitute values for <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> a 1 , a n  , and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi></mrow></annotation-xml></semantics></math> into the formula and simplify.

    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo> </mo><msub/></mrow></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> S n = n( a 1 + a n ) 2 S 10 = 10(5+32) 2 =185
  2. We are given <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics></math> 1 =20 and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics></math> n =−50.

    Use the formula for the general term of an arithmetic sequence to find <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><msub/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> a n = a 1 +(n−1)d −50=20+(n−1)(−5) −70=(n−1)(−5)     14=n−1     15=n

    Substitute values for <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics></math> 1 , a n , n into the formula and simplify.

     

     

    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mtable columnalign="left"><mtr><mtd><mrow/></mtd></mtr><mtr><mtd><msub><mi>S</mi></msub></mtd></mtr></mtable></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> n = n( a 1 + a n ) 2 S 15 = 15(20−50) 2 =−225
  3. To find <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics></math> 1 , substitute <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow></annotation-xml></semantics></math> into the given explicit formula.

    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>a</mi></msub></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> k =3k−8   a 1 =3(1)−8=−5

    We are given that <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mo>=</mo><mn>12.</mn></mrow></annotation-xml></semantics></math> To find <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics></math> 12 , substitute <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>k</mi><mo>=</mo><mn>12</mn></mrow></annotation-xml></semantics></math> into the given explicit formula.

    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext> </mtext><msub/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> a k =3k−8 a 12 =3(12)−8=28

    Substitute values for <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics></math> 1 , a n , and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi></mrow></annotation-xml></semantics></math> into the formula and simplify.

    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext> </mtext><msub/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> S n = n( a 1 + a n ) 2 S 12 = 12(−5+28) 2 =138

Use the formula to find the sum of each arithmetic series.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>1</mtext><mtext>.4 + 1</mtext><mtext>.6 + 1</mtext><mtext>.8 + 2</mtext><mtext>.0 + 2</mtext><mtext>.2 + 2</mtext><mtext>.4 + 2</mtext><mtext>.6 + 2</mtext><mtext>.8 + 3</mtext><mtext>.0 + 3</mtext><mtext>.2 + 3</mtext><mtext>.4</mtext></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>26</mtext><mtext>.4</mtext></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>13 + 21 + 29 + </mtext><mo>…</mo><mtext>+ 69</mtext></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>328</mtext></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics></math> k=1 10 5 −6k

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>−280</mtext></mrow></annotation-xml></semantics></math>

Solving Application Problems with Arithmetic Series

On the Sunday after a minor surgery, a woman is able to walk a half-mile. Each Sunday, she walks an additional quarter-mile. After 8 weeks, what will be the total number of miles she has walked?

This problem can be modeled by an arithmetic series with<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> a 1 = 1 2  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>d</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 4 . We are looking for the total number of miles walked after 8 weeks, so we know that <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mo>=</mo><mn>8</mn><mtext>,</mtext></mrow></annotation-xml></semantics></math> and we are looking for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> S 8 . To find <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics></math> 8 , we can use the explicit formula for an arithmetic sequence.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mtable columnalign="left"><mtr><mtd><mrow/></mtd></mtr><mtr><mtd><msub><mi>a</mi></msub></mtd></mtr></mtable></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> n = a 1 +d(n−1) a 8 = 1 2 + 1 4 (8−1)= 9 4

We can now use the formula for arithmetic series.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mo> </mo><msub/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> S n = n( a 1 + a n ) 2    S 8 = 8( 1 2 + 9 4 ) 2 =11

She will have walked a total of 11 miles.

A man earns $100 in the first week of June. Each week, he earns $12.50 more than the previous week. After 12 weeks, how much has he earned?

$2,025

Using the Formula for Geometric Series

Just as the sum of the terms of an arithmetic sequence is called an arithmetic series, the sum of the terms in a geometric sequence is called a geometric series. Recall that a geometric sequence is a sequence in which the ratio of any two consecutive terms is the common ratio, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>We can write the sum of the first <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi></mrow></annotation-xml></semantics></math> terms of a geometric series as

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> n = a 1 +r a 1 + r 2 a 1 +...+ r n–1 a 1 .

Just as with arithmetic series, we can do some algebraic manipulation to derive a formula for the sum of the first<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>n</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>terms of a geometric series. We will begin by multiplying both sides of the equation by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><msub/></mrow></annotation-xml></semantics></math> S n =r a 1 + r 2 a 1 + r 3 a 1 +...+ r n a 1

Next, we subtract this equation from the original equation.

 

 

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr><mtd><mrow/></mtd></mtr><mtr><mtd><mfrac><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext>    </mtext><msub/></mrow></mtd></mtr></mtable></mrow></mfrac></mtd></mtr></mtable></annotation-xml></semantics></math> S n = a 1 +r a 1 + r 2 a 1 +...+ r n–1 a 1 −r S n =−(r a 1 + r 2 a 1 + r 3 a 1 +...+ r n a 1 ) (1−r) S n = a 1 − r n a 1

Notice that when we subtract, all but the first term of the top equation and the last term of the bottom equation cancel out. To obtain a formula for <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> n , divide both sides by <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>r</mi><mo stretchy="false">)</mo><mo>.</mo></mrow></annotation-xml></semantics></math>

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> n = a 1 (1− r n ) 1−r  r≠1
Formula for the Sum of the First n Terms of a Geometric Series

A geometric series is the sum of the terms in a geometric sequence. The formula for the sum of the first<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>n</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>terms of a geometric sequence is represented as

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> n = a 1 (1− r n ) 1−r  r≠1
Given a geometric series, find the sum of the first n terms.
  1. Identify<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> a 1 , r, and n.
  2. Substitute values for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> a 1 , r, and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics></math> into the formula <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> n = a 1 (1– r n ) 1–r .
  3. Simplify to find <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> n .
Finding the First n Terms of a Geometric Series

Use the formula to find the indicated partial sum of each geometric series.

  1. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> 11for the series<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> 8 + -4 + 2 + </mtext><mo>…</mo></mrow></annotation-xml></semantics></math>
  2. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munderover><mrow><msup><mstyle displaystyle="true" mathsize="75%"><mo>∑</mo></mstyle></msup></mrow></munderover></mrow></annotation-xml></semantics></math> ​ 6 k=1 3⋅ 2 k
  1. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics></math> 1 =8, and we are given that <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mo>=</mo><mn>11.</mn></mrow></annotation-xml></semantics></math>

    We can find <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>r</mi></annotation-xml></semantics></math> by dividing the second term of the series by the first.

    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> −4 8 =− 1 2

    Substitute values for <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics></math> 1 , r, and n into the formula and simplify.

    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>S</mi></msub></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> n = a 1 ( 1− r n ) 1−r S 11 = 8( 1− ( − 1 2 ) 11 ) 1−( − 1 2 ) ≈5.336
  2. Find <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics></math> 1 by substituting <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow></annotation-xml></semantics></math> into the given explicit formula.

    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics></math> 1 =3⋅ 2 1 =6

    We can see from the given explicit formula that <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mn>2.</mn></mrow></annotation-xml></semantics></math> The upper limit of summation is 6, so <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mo>=</mo><mn>6.</mn></mrow></annotation-xml></semantics></math>

    Substitute values for <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics></math> 1 , r, and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics></math> into the formula, and simplify.

    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>S</mi></msub></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> n = a 1 (1− r n ) 1−r S 6 = 6(1− 2 6 ) 1−2 =378

Use the formula to find the indicated partial sum of each geometric series.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> 20 for the series<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> 1,000 + 500 + 250 + </mtext><mo>…</mo></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>≈</mo><mn>2</mn><mo>,</mo><mn>000.00</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics></math> k=1 8 3 k

9,840

Solving an Application Problem with a Geometric Series

At a new job, an employee’s starting salary is $26,750. He receives a 1.6% annual raise. Find his total earnings at the end of 5 years.

The problem can be represented by a geometric series with <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics></math> 1 =26,750; <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mo>=</mo><mn>5</mn><mtext>;</mtext><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>=</mo><mn>1.016.</mn></mrow></annotation-xml></semantics></math> Substitute values for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> a 1 , <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mtext>,</mtext></mrow></annotation-xml></semantics></math> and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi></mrow></annotation-xml></semantics></math>into the formula and simplify to find the total amount earned at the end of 5 years.

 

 

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>S</mi></msub></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> n = a 1 (1− r n ) 1−r S 5 = 26,750(1− 1.016 5 ) 1−1.016 ≈138,099.03

He will have earned a total of $138,099.03 by the end of 5 years.

At a new job, an employee’s starting salary is $32,100. She receives a 2% annual raise. How much will she have earned by the end of 8 years?

$275,513.31

Using the Formula for the Sum of an Infinite Geometric Series

Thus far, we have looked only at finite series. Sometimes, however, we are interested in the sum of the terms of an infinite sequence rather than the sum of only the first <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics></math>terms. An infinite series is the sum of the terms of an infinite sequence. An example of an infinite series is <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>2</mn><mo>+</mo><mn>4</mn><mo>+</mo><mn>6</mn><mo>+</mo><mn>8</mn><mo>+</mo><mn>...</mn></mrow></annotation-xml></semantics></math>

This series can also be written in summation notation as <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics></math> k=1 ∞ 2k, where the upper limit of summation is infinity. Because the terms are not tending to zero, the sum of the series increases without bound as we add more terms. Therefore, the sum of this infinite series is not defined. When the sum is not a real number, we say the series diverges.

Determining Whether the Sum of an Infinite Geometric Series is Defined

If the terms of an infinite geometric series approach 0, the sum of an infinite geometric series can be defined. The terms in this series approach 0:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>1</mn><mo>+</mo><mn>0.2</mn><mo>+</mo><mn>0.04</mn><mo>+</mo><mn>0.008</mn><mo>+</mo><mn>0.0016</mn><mo>+</mo><mn>...</mn></mrow></annotation-xml></semantics></math>

The common ratio <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mtext> = 0</mtext><mtext>.2</mtext><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math> As<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi></mrow></annotation-xml></semantics></math> gets very large, the values of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>r</mi></msup></mrow></annotation-xml></semantics></math> n get very small and approach 0. Each successive term affects the sum less than the preceding term. As each succeeding term gets closer to 0, the sum of the terms approaches a finite value. The terms of any infinite geometric series with <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>1</mn><mo><</mo><mi>r</mi><mo><</mo><mn>1</mn></mrow></annotation-xml></semantics></math> approach 0; the sum of a geometric series is defined when <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>1</mn><mo><</mo><mi>r</mi><mo><</mo><mn>1.</mn></mrow></annotation-xml></semantics></math>

Determining Whether the Sum of an Infinite Geometric Series is Defined

The sum of an infinite series is defined if the series is geometric and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>1</mn><mo><</mo><mi>r</mi><mo><</mo><mn>1.</mn></mrow></annotation-xml></semantics></math>

Given the first several terms of an infinite series, determine if the sum of the series exists.

  1. Find the ratio of the second term to the first term.
  2. Find the ratio of the third term to the second term.
  3. Continue this process to ensure the ratio of a term to the preceding term is constant throughout. If so, the series is geometric.
  4. If a common ratio, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>,</mo></mrow></annotation-xml></semantics></math> was found in step 3, check to see if <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>1</mn><mo><</mo><mi>r</mi><mo><</mo><mn>1</mn></mrow></annotation-xml></semantics></math>. If so, the sum is defined. If not, the sum is not defined.
Determining Whether the Sum of an Infinite Series is Defined

Determine whether the sum of each infinite series is defined.

  1. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>12 + 8 + 4 + </mtext><mo>…</mo></mrow></annotation-xml></semantics></math>
  2. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>3</mn></mfrac></mrow></annotation-xml></semantics></math> 4 + 1 2 + 1 3 +...
  3. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munderover><mstyle displaystyle="true"><mo>∑</mo></mstyle></munderover></mrow></annotation-xml></semantics></math> k=1 ∞ 27⋅ ( 1 3 ) k
  4. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics></math> k=1 ∞ 5k
  1. The ratio of the second term to the first is <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mtext>2</mtext></mfrac></mrow></annotation-xml></semantics></math> 3 , which is not the same as the ratio of the third term to the second, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>1</mn></mfrac></mrow></annotation-xml></semantics></math> 2 .The series is not geometric.
  2. The ratio of the second term to the first is the same as the ratio of the third term to the second. The series is geometric with a common ratio of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>2</mn></mfrac></mrow></annotation-xml></semantics></math> 3 . The sum of the infinite series is defined.

  3. The given formula is exponential with a base of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>1</mn></mfrac></mrow></annotation-xml></semantics></math> 3 ; the series is geometric with a common ratio of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>1</mn></mfrac></mrow></annotation-xml></semantics></math> 3 . The sum of the infinite series is defined.
  4. The given formula is not exponential; the series is not geometric because the terms are increasing, and so cannot yield a finite sum.

Determine whether the sum of the infinite series is defined.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>1</mn></mfrac></mrow></annotation-xml></semantics></math> 3 + 1 2 + 3 4 + 9 8 +...

The sum is defined. It is geometric.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>24</mn><mo>+</mo><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −12 )+6+( −3 )+...

The sum of the infinite series is defined.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munderover><mstyle displaystyle="true" mathsize="140%"><mo>∑</mo></mstyle></munderover></mrow></annotation-xml></semantics></math> k=1 ∞ 15⋅ (–0.3) k

The sum of the infinite series is defined.

Finding Sums of Infinite Series

When the sum of an infinite geometric series exists, we can calculate the sum. The formula for the sum of an infinite series is related to the formula for the sum of the first <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics></math>terms of a geometric series.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> n = a 1 (1− r n ) 1−r

We will examine an infinite series with <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 2 . What happens to <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>r</mi></msup></mrow></annotation-xml></semantics></math> n as <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi></mrow></annotation-xml></semantics></math> increases?

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr><mtd><msup><mrow><mo>(</mo></mrow></msup></mtd></mtr></mtable></annotation-xml></semantics></math> 1 2 ) 2 = 1 4 ( 1 2 ) 3 = 1 8 ( 1 2 ) 4 = 1 16

The value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> r n  decreases rapidly. What happens for greater values of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mo>?</mo></mrow></annotation-xml></semantics></math>

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mrow><mo stretchy="false">(</mo><mfrac/></mrow></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 1 2 ) 10 = 1 1,024 ( 1 2 ) 20 = 1 1,048,576 ( 1 2 ) 30 = 1 1,073,741,824

As <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi></mrow></annotation-xml></semantics></math> gets very large, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>r</mi></msup></mrow></annotation-xml></semantics></math> n gets very small. We say that, as <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi></mrow></annotation-xml></semantics></math> increases without bound, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>r</mi></msup></mrow></annotation-xml></semantics></math> napproaches 0. As <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>r</mi></msup></mrow></annotation-xml></semantics></math> n approaches 0,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>1</mn><mo>−</mo><msup/></mrow></annotation-xml></semantics></math> r n approaches 1. When this happens, the numerator approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> a 1 . This give us a formula for the sum of an infinite geometric series.

Formula for the Sum of an Infinite Geometric Series

The formula for the sum of an infinite geometric series with <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>−1</mn><mo><</mo><mi>r</mi><mo><</mo><mn>1</mn></mrow></annotation-xml></semantics></math> is

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>S</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> a 1 1−r

Given an infinite geometric series, find its sum.

  1. Identify<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics></math> 1and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>.</mo></mrow></annotation-xml></semantics></math>
  2. Confirm that <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>–</mo><mn>1</mn><mo><</mo><mi>r</mi><mo><</mo><mn>1.</mn></mrow></annotation-xml></semantics></math>
  3. Substitute values for <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics></math> 1 and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>r</mi></annotation-xml></semantics></math> into the formula, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>S</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> a 1 1−r .
  4. Simplify to find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>S</mi><mo>.</mo></mrow></annotation-xml></semantics></math>
Finding the Sum of an Infinite Geometric Series

Find the sum, if it exists, for the following:

  1. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>10</mn><mo>+</mo><mn>9</mn><mo>+</mo><mn>8</mn><mo>+</mo><mn>7</mn><mo>+</mo><mo>…</mo></mrow></annotation-xml></semantics></math>
  2. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>248.6</mn><mo>+</mo><mn>99.44</mn><mo>+</mo><mn>39.776</mn><mo>+</mo><mtext> </mtext><mo>…</mo></mrow></annotation-xml></semantics></math>
  3. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munderover><mstyle displaystyle="true"><mo>∑</mo></mstyle></munderover></mrow></annotation-xml></semantics></math> k=1 ∞ 4,374⋅ (– 1 3 ) k–1
  4. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munderover><mstyle displaystyle="true"><mo>∑</mo></mstyle></munderover></mrow></annotation-xml></semantics></math> k=1 ∞ 1 9 ⋅ ( 4 3 ) k
  1. There is not a constant ratio; the series is not geometric.
  2. There is a constant ratio; the series is geometric. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics></math> 1 =248.6and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 99.44 248.6 =0.4, so the sum exists. Substitute <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics></math> 1 =248.6 and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mn>0.4</mn></mrow></annotation-xml></semantics></math> into the formula and simplify to find the sum:

    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>S</mi><mo>=</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> a 1 1−r S= 248.6 1−0.4 =414. 3 ¯
  3. The formula is exponential, so the series is geometric with <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mo>–</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 3 . Find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics></math> 1 by substituting <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow></annotation-xml></semantics></math> into the given explicit formula:

    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics></math> 1 =4,374⋅ (– 1 3 ) 1–1 =4,374

    Substitute <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics></math> 1 =4,374 and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 3 into the formula, and simplify to find the sum:

    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>S</mi><mo>=</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> a 1 1−r S= 4,374 1−(− 1 3 ) =3,280.5
  4. The formula is exponential, so the series is geometric, but<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>></mo><mn>1.</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>The sum does not exist.
Finding an Equivalent Fraction for a Repeating Decimal

Find an equivalent fraction for the repeating decimal <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mn>0.</mn><mover accent="true"><mn>3</mn><mo>¯</mo></mover></annotation-xml></semantics></math>

We notice the repeating decimal <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mn>0.</mn><mover accent="true"><mn>3</mn><mo>¯</mo></mover><mo>=</mo><mn>0.333...</mn></annotation-xml></semantics></math> so we can rewrite the repeating decimal as a sum of terms.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mn>0.</mn><mover accent="true"><mn>3</mn><mo>¯</mo></mover><mo>=</mo><mn>0.3</mn><mo>+</mo><mn>0.03</mn><mo>+</mo><mn>0.003</mn><mo>+</mo><mn>...</mn></annotation-xml></semantics></math>

Looking for a pattern, we rewrite the sum, noticing that we see the first term multiplied to 0.1 in the second term, and the second term multiplied to 0.1 in the third term.

...

Notice the pattern; we multiply each consecutive term by a common ratio of 0.1 starting with the first term of 0.3. So, substituting into our formula for an infinite geometric sum, we have

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> n = a 1 1−r = 0.3 1−0.1 = 0.3 0.9 = 1 3 .

Find the sum, if it exists.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>2</mn><mo>+</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 3 + 2 9 +...

3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics></math> k=1 ∞ 0.76k+1

The series is not geometric.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics></math> k=1 ∞ ( − 3 8 ) k

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 11

Solving Annuity Problems

At the beginning of the section, we looked at a problem in which a couple invested a set amount of money each month into a college fund for six years. An annuity is an investment in which the purchaser makes a sequence of periodic, equal payments. To find the amount of an annuity, we need to find the sum of all the payments and the interest earned. In the example, the couple invests $50 each month. This is the value of the initial deposit. The account paid 6% annual interest, compounded monthly. To find the interest rate per payment period, we need to divide the 6% annual percentage interest (APR) rate by 12. So the monthly interest rate is 0.5%. We can multiply the amount in the account each month by 100.5% to find the value of the account after interest has been added.

We can find the value of the annuity right after the last deposit by using a geometric series with <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics></math> 1 =50 and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mn>100.5</mn><mi>%</mi><mo>=</mo><mn>1.005.</mn></mrow></annotation-xml></semantics></math> After the first deposit, the value of the annuity will be $50. Let us see if we can determine the amount in the college fund and the interest earned.

We can find the value of the annuity after <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics></math> deposits using the formula for the sum of the first <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics></math> terms of a geometric series. In 6 years, there are 72 months, so <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mo>=</mo><mn>72.</mn></mrow></annotation-xml></semantics></math> We can substitute <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics></math> 1 =50, r=1.005, and n=72 into the formula, and simplify to find the value of the annuity after 6 years.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> 72 = 50(1− 1.005 72 ) 1−1.005 ≈4,320.44

After the last deposit, the couple will have a total of $4,320.44 in the account. Notice, the couple made 72 payments of $50 each for a total of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>72(50) = $3,600</mtext><mtext>.</mtext><mtext> </mtext></mrow></annotation-xml></semantics></math>This means that because of the annuity, the couple earned $720.44 interest in their college fund.

Given an initial deposit and an interest rate, find the value of an annuity.

  1. Determine<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> a 1 , the value of the initial deposit.
  2. Determine<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>n</mi><mtext>,</mtext><mtext> </mtext></mrow></annotation-xml></semantics></math>the number of deposits.
  3. Determine<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>.</mo></mrow></annotation-xml></semantics></math>
    1. Divide the annual interest rate by the number of times per year that interest is compounded.
    2. Add 1 to this amount to find <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>.</mo></mrow></annotation-xml></semantics></math>
  4. Substitute values for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> a 1 , r, and n  into the formula for the sum of the first <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics></math> terms of a geometric series,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> n = a 1 (1– r n ) 1–r .
  5. Simplify to find <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> n , the value of the annuity after <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics></math> deposits.
Solving an Annuity Problem

A deposit of $100 is placed into a college fund at the beginning of every month for 10 years. The fund earns 9% annual interest, compounded monthly, and paid at the end of the month. How much is in the account right after the last deposit?

The value of the initial deposit is $100, so<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> a 1 =100. A total of 120 monthly deposits are made in the 10 years, so<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mo>=</mo><mn>120.</mn></mrow></annotation-xml></semantics></math> To find <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>divide the annual interest rate by 12 to find the monthly interest rate and add 1 to represent the new monthly deposit.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mn>1</mn><mo>+</mo><mfrac/></mrow></annotation-xml></semantics></math> 0.09 12 =1.0075

Substitute<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> a 1 =100, r=1.0075, and n=120 into the formula for the sum of the first <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics></math> terms of a geometric series, and simplify to find the value of the annuity.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> 120 = 100(1− 1.0075 120 ) 1−1.0075 ≈19,351.43

So the account has $19,351.43 after the last deposit is made.

At the beginning of each month, $200 is deposited into a retirement fund. The fund earns 6% annual interest, compounded monthly, and paid into the account at the end of the month. How much is in the account if deposits are made for 10 years?

$92,408.18

Access these online resources for additional instruction and practice with series.

Key Equations

sum of the first<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>n</mi><mtext> </mtext></mrow></annotation-xml></semantics></math> terms of an arithmetic series <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> n = n( a 1 + a n ) 2
sum of the first<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>n</mi><mtext> </mtext></mrow></annotation-xml></semantics></math> terms of a geometric series <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> n = a 1 (1− r n ) 1−r ⋅r≠1
sum of an infinite geometric series with<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo>–</mo><mn>1</mn><mo><</mo><mi>r</mi><mo><</mo><mtext> </mtext><mn>1</mn></mrow></annotation-xml></semantics></math> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> n = a 1 1−r ⋅r≠1

Key Concepts

  • The sum of the terms in a sequence is called a series.
  • A common notation for series is called summation notation, which uses the Greek letter sigma to represent the sum. See [link].
  • The sum of the terms in an arithmetic sequence is called an arithmetic series.
  • The sum of the first<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics></math>terms of an arithmetic series can be found using a formula. See [link] and [link].
  • The sum of the terms in a geometric sequence is called a geometric series.
  • The sum of the first<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics></math>terms of a geometric series can be found using a formula. See [link] and [link].
  • The sum of an infinite series exists if the series is geometric with <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>–1</mn><mo><</mo><mi>r</mi><mo><</mo><mn>1.</mn></mrow></annotation-xml></semantics></math>
  • If the sum of an infinite series exists, it can be found using a formula. See [link], [link], and [link].
  • An annuity is an account into which the investor makes a series of regularly scheduled payments. The value of an annuity can be found using geometric series. See [link].

Section Exercises

Verbal

What is an <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mtext>th</mtext></mrow></annotation-xml></semantics></math> partial sum?

An <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mtext>th</mtext></mrow></annotation-xml></semantics></math> partial sum is the sum of the first <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics></math> terms of a sequence.

What is the difference between an arithmetic sequence and an arithmetic series?

What is a geometric series?

A geometric series is the sum of the terms in a geometric sequence.

How is finding the sum of an infinite geometric series different from finding the <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mtext>th</mtext></mrow></annotation-xml></semantics></math> partial sum?

What is an annuity?

An annuity is a series of regular equal payments that earn a constant compounded interest.

Algebraic

For the following exercises, express each description of a sum using summation notation.

The sum of terms <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>m</mi></msup></mrow></annotation-xml></semantics></math> 2 +3m from <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow></annotation-xml></semantics></math> to <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>m</mi><mo>=</mo><mn>5</mn></mrow></annotation-xml></semantics></math>

The sum from of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow></annotation-xml></semantics></math> to <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mo>=</mo><mn>4</mn></mrow></annotation-xml></semantics></math> of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>5</mn><mi>n</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics></math> n=0 4 5n

The sum of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>6</mn><mi>k</mi><mo>−</mo><mn>5</mn></mrow></annotation-xml></semantics></math> from <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>k</mi><mo>=</mo><mo>−</mo><mn>2</mn></mrow></annotation-xml></semantics></math> to <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow></annotation-xml></semantics></math>

The sum that results from adding the number 4 five times

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics></math> k=1 5 4

For the following exercises, express each arithmetic sum using summation notation.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>5</mn><mo>+</mo><mn>10</mn><mo>+</mo><mn>15</mn><mo>+</mo><mn>20</mn><mo>+</mo><mn>25</mn><mo>+</mo><mn>30</mn><mo>+</mo><mn>35</mn><mo>+</mo><mn>40</mn><mo>+</mo><mn>45</mn><mo>+</mo><mn>50</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>10</mn><mo>+</mo><mn>18</mn><mo>+</mo><mn>26</mn><mo>+</mo><mo>…</mo><mo>+</mo><mn>162</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics></math> k=1 20 8k+2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>1</mn></mfrac></mrow></annotation-xml></semantics></math> 2 +1+ 3 2 +2+…+4

For the following exercises, use the formula for the sum of the first <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics></math> terms of each arithmetic sequence.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>3</mn></mfrac></mrow></annotation-xml></semantics></math> 2 +2+ 5 2 +3+ 7 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> 5 = 5( 3 2 + 7 2 ) 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>19</mn><mo>+</mo><mn>25</mn><mo>+</mo><mn>31</mn><mo>+</mo><mo>…</mo><mo>+</mo><mn>73</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>3.2</mn><mo>+</mo><mn>3.4</mn><mo>+</mo><mn>3.6</mn><mo>+</mo><mo>…</mo><mo>+</mo><mn>5.6</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> 13 = 13( 3.2+5.6 ) 2

For the following exercises, express each geometric sum using summation notation.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>1</mn><mo>+</mo><mn>3</mn><mo>+</mo><mn>9</mn><mo>+</mo><mn>27</mn><mo>+</mo><mn>81</mn><mo>+</mo><mn>243</mn><mo>+</mo><mn>729</mn><mo>+</mo><mn>2187</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>8</mn><mo>+</mo><mn>4</mn><mo>+</mo><mn>2</mn><mo>+</mo><mo>…</mo><mo>+</mo><mn>0.125</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics></math> k=1 7 8⋅ 0.5 k−1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 6 + 1 12 − 1 24 +…+ 1 768

For the following exercises, use the formula for the sum of the first <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi></mrow></annotation-xml></semantics></math> terms of each geometric sequence, and then state the indicated sum.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>9</mn><mo>+</mo><mn>3</mn><mo>+</mo><mn>1</mn><mo>+</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 3 + 1 9

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> 5 = 9( 1− ( 1 3 ) 5 ) 1− 1 3 = 121 9 ≈13.44

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics></math> n=1 9 5⋅ 2 n−1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics></math> a=1 11 64⋅ 0.2 a−1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> 11 = 64( 1− 0.2 11 ) 1−0.2 = 781,249,984 9,765,625 ≈80

For the following exercises, determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>12</mn><mo>+</mo><mn>18</mn><mo>+</mo><mn>24</mn><mo>+</mo><mn>30</mn><mo>+</mo><mn>...</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>2</mn><mo>+</mo><mn>1.6</mn><mo>+</mo><mn>1.28</mn><mo>+</mo><mn>1.024</mn><mo>+</mo><mn>...</mn></mrow></annotation-xml></semantics></math>

The series is defined. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>S</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 1−0.8

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics></math> m=1 ∞ 4 m−1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munderover><mrow><msup><mstyle displaystyle="true" mathsize="140%"><mo>∑</mo></mstyle></msup></mrow></munderover></mrow></annotation-xml></semantics></math> ​ ∞ k=1 − ( − 1 2 ) k−1

The series is defined. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>S</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> −1 1−( − 1 2 )

Graphical

For the following exercises, use the following scenario. Javier makes monthly deposits into a savings account. He opened the account with an initial deposit of $50. Each month thereafter he increased the previous deposit amount by $20.

Graph the arithmetic sequence showing one year of Javier’s deposits.

Graph the arithmetic series showing the monthly sums of one year of Javier’s deposits.

Graph of Javier's deposits where the x-axis is the months of the year and the y-axis is the sum of deposits.

For the following exercises, use the geometric series<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></msup></mrow></annotation-xml></semantics></math> k=1 ∞ ( 1 2 ) k .

Graph the first 7 partial sums of the series.

What number does <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> n seem to be approaching in the graph? Find the sum to explain why this makes sense.

Sample answer: The graph of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> n seems to be approaching 1. This makes sense because<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics></math> k=1 ∞ ( 1 2 ) kis a defined infinite geometric series with <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>S</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 2 1–( 1 2 ) =1.

Numeric

For the following exercises, find the indicated sum.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics></math> a=1 14 a

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics></math> n=1 6 n(n−2)

49

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics></math> k=1 17 k 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics></math> k=1 7 2 k

254

For the following exercises, use the formula for the sum of the first <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics></math> terms of an arithmetic series to find the sum.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>1.7</mn><mo>+</mo><mo>−</mo><mn>0.4</mn><mo>+</mo><mn>0.9</mn><mo>+</mo><mn>2.2</mn><mo>+</mo><mn>3.5</mn><mo>+</mo><mn>4.8</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>6</mn><mo>+</mo><mfrac/></mrow></annotation-xml></semantics></math> 15 2 +9+ 21 2 +12+ 27 2 +15

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> 7 = 147 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>1</mn><mo>+</mo><mn>3</mn><mo>+</mo><mn>7</mn><mo>+</mo><mn>...</mn><mo>+</mo><mn>31</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics></math> k=1 11 ( k 2 − 1 2 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> 11 = 55 2

For the following exercises, use the formula for the sum of the first <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics></math> terms of a geometric series to find the partial sum.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> 6 for the series <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>2</mn><mo>−</mo><mn>10</mn><mo>−</mo><mn>50</mn><mo>−</mo><mn>250...</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> 7 for the series <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>0.4</mn><mo>−</mo><mn>2</mn><mo>+</mo><mn>10</mn><mo>−</mo><mn>50...</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> 7 =5208.4

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics></math> k=1 9 2 k−1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics></math> n=1 10 −2⋅ ( 1 2 ) n−1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>S</mi></msub></mrow></annotation-xml></semantics></math> 10 =− 1023 256

For the following exercises, find the sum of the infinite geometric series.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>4</mn><mo>+</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 2 ...

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>1</mn><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 4 − 1 16 − 1 64 ...

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>S</mi><mo>=</mo><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 4 3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munderover><mrow><msup><mstyle displaystyle="true" mathsize="140%"><mo>∑</mo></mstyle></msup></mrow></munderover></mrow></annotation-xml></semantics></math> ​ ∞ k=1 3⋅ ( 1 4 ) k−1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics></math> n=1 ∞ 4.6⋅ 0.5 n−1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>S</mi><mo>=</mo><mn>9.2</mn></mrow></annotation-xml></semantics></math>

For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate.

Deposit amount: <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>$</mtext><mn>50</mn><mo>;</mo></mrow></annotation-xml></semantics></math> total deposits: <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>60</mn><mo>;</mo></mrow></annotation-xml></semantics></math> interest rate: <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>5</mn><mi>%</mi><mo>,</mo></mrow></annotation-xml></semantics></math> compounded monthly

Deposit amount: <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>$</mtext><mn>150</mn><mo>;</mo></mrow></annotation-xml></semantics></math> total deposits: <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>24</mn><mo>;</mo></mrow></annotation-xml></semantics></math> interest rate: <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>3</mn><mi>%</mi><mo>,</mo></mrow></annotation-xml></semantics></math> compounded monthly

$3,705.42

Deposit amount: <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>$</mtext><mn>450</mn><mo>;</mo></mrow></annotation-xml></semantics></math> total deposits: <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>60</mn><mo>;</mo></mrow></annotation-xml></semantics></math> interest rate: <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>4.5</mn><mi>%</mi><mo>,</mo></mrow></annotation-xml></semantics></math> compounded quarterly

Deposit amount: <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>$</mtext><mn>100</mn><mo>;</mo></mrow></annotation-xml></semantics></math> total deposits: <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>120</mn><mo>;</mo></mrow></annotation-xml></semantics></math> interest rate: <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>10</mn><mi>%</mi><mo>,</mo></mrow></annotation-xml></semantics></math> compounded semi-annually

$695,823.97

Extensions

The sum of terms <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>50</mn><mo>−</mo><msup/></mrow></annotation-xml></semantics></math> k 2 from <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>k</mi><mo>=</mo><mi>x</mi></mrow></annotation-xml></semantics></math> through <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mn>7</mn></annotation-xml></semantics></math> is <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>115.</mn></mrow></annotation-xml></semantics></math> What is x?

Write an explicit formula for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics></math> ksuch that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics></math> k=0 6 a k =189. Assume this is an arithmetic series.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>a</mi></msub></mrow></annotation-xml></semantics></math> k =30−k

Find the smallest value of n such that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo></munderover></mstyle></mrow></annotation-xml></semantics></math> k=1 n (3k–5)>100.

How many terms must be added before the series <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>1</mn><mo>−</mo><mn>3</mn><mo>−</mo><mn>5</mn><mo>−</mo><mn>7</mn><mn>....</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>has a sum less than <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>75</mn><mo>?</mo></mrow></annotation-xml></semantics></math>

9 terms

Write <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>0.</mn><mover accent="true"/></mrow></annotation-xml></semantics></math> 65 ¯ as an infinite geometric series using summation notation. Then use the formula for finding the sum of an infinite geometric series to convert <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>0.</mn><mover accent="true"/></mrow></annotation-xml></semantics></math> 65 ¯ to a fraction.

The sum of an infinite geometric series is five times the value of the first term. What is the common ratio of the series?

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 4 5

To get the best loan rates available, the Riches want to save enough money to place 20% down on a $160,000 home. They plan to make monthly deposits of $125 in an investment account that offers 8.5% annual interest compounded semi-annually. Will the Riches have enough for a 20% down payment after five years of saving? How much money will they have saved?

Karl has two years to save <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>$</mi><mn>10</mn><mo>,</mo><mn>000</mn></mrow></annotation-xml></semantics></math> to buy a used car when he graduates. To the nearest dollar, what would his monthly deposits need to be if he invests in an account offering a 4.2% annual interest rate that compounds monthly?

$400 per month

Real-World Applications

Keisha devised a week-long study plan to prepare for finals. On the first day, she plans to study for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>1</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>hour, and each successive day she will increase her study time by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>30</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>minutes. How many hours will Keisha have studied after one week?

A boulder rolled down a mountain, traveling 6 feet in the first second. Each successive second, its distance increased by 8 feet. How far did the boulder travel after 10 seconds?

420 feet

A scientist places 50 cells in a petri dish. Every hour, the population increases by 1.5%. What will the cell count be after 1 day?

A pendulum travels a distance of 3 feet on its first swing. On each successive swing, it travels <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>3</mn></mfrac></mrow></annotation-xml></semantics></math> 4 the distance of the previous swing. What is the total distance traveled by the pendulum when it stops swinging?

12 feet

Rachael deposits $1,500 into a retirement fund each year. The fund earns 8.2% annual interest, compounded monthly. If she opened her account when she was 19 years old, how much will she have by the time she is 55? How much of that amount will be interest earned?

Glossary

annuity
an investment in which the purchaser makes a sequence of periodic, equal payments
arithmetic series
the sum of the terms in an arithmetic sequence
diverge
a series is said to diverge if the sum is not a real number
geometric series
the sum of the terms in a geometric sequence
index of summation
in summation notation, the variable used in the explicit formula for the terms of a series and written below the sigma with the lower limit of summation
infinite series
the sum of the terms in an infinite sequence
lower limit of summation
the number used in the explicit formula to find the first term in a series
nth partial sum
the sum of the first<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics></math>terms of a sequence
series
the sum of the terms in a sequence
summation notation
a notation for series using the Greek letter sigma; it includes an explicit formula and specifies the first and last terms in the series
upper limit of summation
the number used in the explicit formula to find the last term in a series