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Mathematics LibreTexts

11-2. Arithmetic Sequences

Arithmetic Sequences
In this section, you will:
  • Find the common difference for an arithmetic sequence.
  • Write terms of an arithmetic sequence.
  • Use a recursive formula for an arithmetic sequence.
  • Use an explicit formula for an arithmetic sequence.

Companies often make large purchases, such as computers and vehicles, for business use. The book-value of these supplies decreases each year for tax purposes. This decrease in value is called depreciation. One method of calculating depreciation is straight-line depreciation, in which the value of the asset decreases by the same amount each year.

As an example, consider a woman who starts a small contracting business. She purchases a new truck for $25,000. After five years, she estimates that she will be able to sell the truck for $8,000. The loss in value of the truck will therefore be $17,000, which is $3,400 per year for five years. The truck will be worth $21,600 after the first year; $18,200 after two years; $14,800 after three years; $11,400 after four years; and $8,000 at the end of five years. In this section, we will consider specific kinds of sequences that will allow us to calculate depreciation, such as the truck’s value.

Finding Common Differences

The values of the truck in the example are said to form an arithmetic sequence because they change by a constant amount each year. Each term increases or decreases by the same constant value called the common difference of the sequence. For this sequence, the common difference is –3,400.

A sequence, {25000, 21600, 18200, 14800, 8000}, that shows the terms differ only by -3400.

The sequence below is another example of an arithmetic sequence. In this case, the constant difference is 3. You can choose any term of the sequence, and add 3 to find the subsequent term.

A sequence {3, 6, 9, 12, 15, ...} that shows the terms only differ by 3.
Arithmetic Sequence

An arithmetic sequence is a sequence that has the property that the difference between any two consecutive terms is a constant. This constant is called the common difference. If <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 is the first term of an arithmetic sequence and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>d</mi></annotation-xml></semantics></math> is the common difference, the sequence will be:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>{</mo><msub/></mrow></annotation-xml></semantics></math> a n }={ a 1 , a 1 +d, a 1 +2d, a 1 +3d,...}
Finding Common Differences

Is each sequence arithmetic? If so, find the common difference.

  1. <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>2</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>8</mn><mo>,</mo><mn>16</mn><mo>,</mo><mn>...</mn><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><mo>{</mo><mo>−</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>9</mn><mo>,</mo><mn>13</mn><mo>,</mo><mn>...</mn><mo>}</mo></mrow></annotation-xml></semantics></math>

Subtract each term from the subsequent term to determine whether a common difference exists.

  1. The sequence is not arithmetic because there is no common difference.

  2. The sequence is arithmetic because there is a common difference. The common difference is 4.

Analysis

The graph of each of these sequences is shown in [link]. We can see from the graphs that, although both sequences show growth, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>a</mi></annotation-xml></semantics></math> is not linear whereas <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>b</mi></annotation-xml></semantics></math> is linear. Arithmetic sequences have a constant rate of change so their graphs will always be points on a line.

<figure id="CNX_Precalculus_Figure_11_02_003">Two graphs of arithmetic sequences. Graph (a) grows exponentially while graph (b) grows linearly.</figure>

If we are told that a sequence is arithmetic, do we have to subtract every term from the following term to find the common difference?

No. If we know that the sequence is arithmetic, we can choose any one term in the sequence, and subtract it from the subsequent term to find the common difference.

Is the given sequence arithmetic? If so, find the common difference.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>{</mo><mn>18</mn><mo>,</mo><mtext> </mtext><mn>16</mn><mo>,</mo><mtext> </mtext><mn>14</mn><mo>,</mo><mtext> </mtext><mn>12</mn><mo>,</mo><mtext> </mtext><mn>10</mn><mo>,</mo><mo>…</mo><mo>}</mo></mrow></annotation-xml></semantics></math>

The sequence is arithmetic. The common difference is <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>–</mo><mn>2.</mn></mrow></annotation-xml></semantics></math>

Is the given sequence arithmetic? If so, find the common difference.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mtext> </mtext><mn>3</mn><mo>,</mo><mtext> </mtext><mn>6</mn><mo>,</mo><mtext> </mtext><mn>10</mn><mo>,</mo><mtext> </mtext><mn>15</mn><mo>,</mo><mo>…</mo><mo>}</mo></mrow></annotation-xml></semantics></math>

The sequence is not arithmetic because <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>3</mn><mo>−</mo><mn>1</mn><mo>≠</mo><mn>6</mn><mo>−</mo><mn>3.</mn></mrow></annotation-xml></semantics></math>

Writing Terms of Arithmetic Sequences

Now that we can recognize an arithmetic sequence, we will find the terms if we are given the first term and the common difference. The terms can be found by beginning with the first term and adding the common difference repeatedly. In addition, any term can also be found by plugging in the values of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></annotation-xml></semantics></math> and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>d</mi></annotation-xml></semantics></math> into formula below.

<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> n = a 1 +(n−1)d

Given the first term and the common difference of an arithmetic sequence, find the first several terms.

  1. Add the common difference to the first term to find the second term.
  2. Add the common difference to the second term to find the third term.
  3. Continue until all of the desired terms are identified.
  4. Write the terms separated by commas within brackets.
Writing Terms of Arithmetic Sequences

Write the first five terms of the arithmetic sequence 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 =17 and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>d</mi><mo>=</mo><mo>−</mo><mn>3</mn></mrow></annotation-xml></semantics></math>.

Adding<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo>−</mo><mn>3</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>is the same as subtracting 3. Beginning with the first term, subtract 3 from each term to find the next term.

The first five terms are <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo>{</mo><mn>17</mn><mo>,</mo><mtext> </mtext><mn>14</mn><mo>,</mo><mtext> </mtext><mn>11</mn><mo>,</mo><mtext> </mtext><mn>8</mn><mo>,</mo><mtext> </mtext><mn>5</mn><mo>}</mo></mrow></annotation-xml></semantics></math>

Analysis

As expected, the graph of the sequence consists of points on a line as shown in [link].

<figure id="CNX_Precalculus_Figure_11_02_004">Graph of the arithmetic sequence. The points form a negative line.</figure>

List the first five terms of the arithmetic sequence 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 =1 and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>d</mi><mo>=</mo><mn>5</mn></mrow></annotation-xml></semantics></math>.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>{</mo></mrow></mrow></annotation-xml></semantics></math> 1 ,   6 ,   11 ,   16 ,   21 }

Given any the first term and any other term in an arithmetic sequence, find a given term.

  1. Substitute the values given 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<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 = a 1 +(n−1)d to solve for<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></mrow></annotation-xml></semantics></math>
  2. Find a given term by substituting the appropriate 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 ,n, and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>d</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>a</mi></msub></mrow></annotation-xml></semantics></math> n = a 1 +(n−1)d.
Writing Terms of Arithmetic Sequences

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 =8 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> 4 =14, 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> 5.

The sequence can be written in terms of the initial term 8 and the common difference <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>d</mi></annotation-xml></semantics></math>.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>{</mo></mrow></mrow></annotation-xml></semantics></math> 8,8+d,8+2d,8+3d }

We know the fourth term equals 14; we know the fourth term has the form <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 +3d=8+3d.

We can find the common difference <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>d</mi></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><msub><mi>a</mi></msub></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> n = a 1 +(n−1)d a 4 = a 1 +3d a 4 =8+3d Write the fourth term of the sequence in terms of  a 1  and d. 14=8+3dSubstitute 14 for  a 4 .    d=2 Solve for the common difference.

Find the fifth term by adding the common difference to the fourth term.

<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> 5 = a 4 +2=16
Analysis

Notice that the common difference is added to the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by adding the common difference to the first term nine times or by using the equation <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 = a 1 +( n−1 )d.

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> 3 =7 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> 5 =17, 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> 2.

<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> 2 = 2

Using Recursive Formulas for Arithmetic Sequences

Some arithmetic sequences are defined in terms of the previous term using a recursive formula. The formula provides an algebraic rule for determining the terms of the sequence. A recursive formula allows us to find any term of an arithmetic sequence using a function of the preceding term. Each term is the sum of the previous term and the common difference. For example, if the common difference is 5, then each term is the previous term plus 5. As with any recursive formula, the first term must be given.

<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> n = a n−1 +d n≥2
Recursive Formula for an Arithmetic Sequence

The recursive formula for an arithmetic sequence with common difference <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>d</mi></annotation-xml></semantics></math> is:

<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> n = a n−1 +d n≥2

Given an arithmetic sequence, write its recursive formula.

  1. Subtract any term from the subsequent term to find the common difference.
  2. State the initial term and substitute the common difference into the recursive formula for arithmetic sequences.
Writing a Recursive Formula for an Arithmetic Sequence

Write a recursive formula for the arithmetic sequence.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>{</mo><mo>−</mo><mn>18</mn><mtext>, </mtext><mo>−</mo><mn>7</mn><mtext>, </mtext><mn>4</mn><mtext>, </mtext><mn>15</mn><mtext>, </mtext><mn>26</mn><mtext>, …</mtext><mo>}</mo></mrow></annotation-xml></semantics></math>

The first term is given as <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>−18</mn></mrow></annotation-xml></semantics></math>. The common difference can be found by subtracting the first term from the second term.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>d</mi><mo>=</mo><mn>−7</mn><mo>−</mo><mo stretchy="false">(</mo><mn>−18</mn><mo stretchy="false">)</mo><mo>=</mo><mn>11</mn></mrow></annotation-xml></semantics></math>

Substitute the initial term and the common difference into the recursive formula for arithmetic sequences.

<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> 1 =−18 a n = a n−1 +11, for n≥2
Analysis

We see that the common difference is the slope of the line formed when we graph the terms of the sequence, as shown in[link]. The growth pattern of the sequence shows the constant difference of 11 units.

<figure id="CNX_Precalculus_Figure_11_02_005">Graph of the arithmetic sequence. The points form a positive line.</figure>

Do we have to subtract the first term from the second term to find the common difference?

No. We can subtract any term in the sequence from the subsequent term. It is, however, most common to subtract the first term from the second term because it is often the easiest method of finding the common difference.

Write a recursive formula for the arithmetic sequence.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>{</mo><mn>25</mn><mtext>, </mtext><mo> </mo><mn>37</mn><mtext>, </mtext><mo> </mo><mn>49</mn><mtext>, </mtext><mo> </mo><mn>61</mn><mtext>, </mtext><mo> </mo><mtext>…</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><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>a</mi></msub></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 1 = 25 a n = a n − 1 + 12 ,  for  n ≥ 2

Using Explicit Formulas for Arithmetic Sequences

We can think of an arithmetic sequence as a function on the domain of the natural numbers; it is a linear function because it has a constant rate of change. The common difference is the constant rate of change, or the slope of the function. We can construct the linear function if we know the slope and the vertical intercept.

<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> n = a 1 + d ( n − 1 )

To find the y-intercept of the function, we can subtract the common difference from the first term of the sequence. Consider the following sequence.

A sequence, {200, 150, 100, 50, 0, ...}, that shows the terms differ only by -50.

The common difference is <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>50</mn></mrow></annotation-xml></semantics></math>, so the sequence represents a linear function with a slope of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>50</mn></mrow></annotation-xml></semantics></math>. To find the <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>y</mi></annotation-xml></semantics></math>-intercept, we subtract <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>50</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><mn>200</mn><mo>:</mo><mtext> </mtext><mn>200</mn><mo>−</mo><mo stretchy="false">(</mo><mo>−</mo><mn>50</mn><mo stretchy="false">)</mo><mo>=</mo><mn>200</mn><mo>+</mo><mn>50</mn><mo>=</mo><mn>250</mn></mrow></annotation-xml></semantics></math>. You can also find the <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>y</mi></annotation-xml></semantics></math>-intercept by graphing the function and determining where a line that connects the points would intersect the vertical axis. The graph is shown in [link].

<figure id="CNX_Precalculus_Figure_11_02_007">Graph of the arithmetic sequence. The points form a negative line.</figure>

Recall the slope-intercept form of a line is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>b</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>When dealing with sequences, we use <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 in place of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi></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> in place of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>If we know the slope and vertical intercept of the function, we can substitute them for <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>m</mi></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>b</mi></mrow></annotation-xml></semantics></math> in the slope-intercept form of a line. Substituting<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo>−</mo><mn>50</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>for the slope and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>250</mn></mrow></annotation-xml></semantics></math> for the vertical intercept, we get the following equation:

<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> n =−50n+250

We do not need to find the vertical intercept to write an explicit formula for an arithmetic sequence. Another explicit formula for this sequence is <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 =200−50(n−1), which simplifies to<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 =−50n+250.

Explicit Formula for an Arithmetic Sequence

An explicit formula for 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> term of an arithmetic sequence is given by

<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> n = a 1 +d(n−1)

Given the first several terms for an arithmetic sequence, write an explicit formula.

  1. Find the common difference, <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> 2 − a 1 .
  2. Substitute the common difference and the first term into <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 = a 1 +d(n−1).
Writing the nth Term Explicit Formula for an Arithmetic Sequence

Write an explicit formula for the arithmetic sequence.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>{</mo><mn>2</mn><mtext>, </mtext><mn>12</mn><mtext>, </mtext><mn>22</mn><mtext>, </mtext><mn>32</mn><mtext>, </mtext><mn>42</mn><mtext>, …</mtext><mo>}</mo></mrow></annotation-xml></semantics></math>

The common difference can be found by subtracting the first term from the second term.

<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"><mi>d</mi></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> = a 2 − a 1 =12−2 =10

The common difference is 10. Substitute the common difference and the first term of the sequence 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>a</mi></msub></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> n =2+10(n−1) a n =10n−8
Analysis

The graph of this sequence, represented in [link], shows a slope of 10 and a vertical intercept of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>8</mn></mrow></annotation-xml></semantics></math>.

<figure id="CNX_Precalculus_Figure_11_02_008">Graph of the arithmetic sequence. The points form a positive line.</figure>

Write an explicit formula for the following arithmetic sequence.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>{</mo><mn>50</mn><mo>,</mo><mn>47</mn><mo>,</mo><mn>44</mn><mo>,</mo><mn>41</mn><mo>,</mo><mo>…</mo><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><msub><mi>a</mi></msub></mrow></annotation-xml></semantics></math> n = 53 − 3 n

Finding the Number of Terms in a Finite Arithmetic Sequence

Explicit formulas can be used to determine the number of terms in a finite arithmetic sequence. We need to find the common difference, and then determine how many times the common difference must be added to the first term to obtain the final term of the sequence.

Given the first three terms and the last term of a finite arithmetic sequence, find the total number of terms.

  1. Find 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>
  2. Substitute the common difference and the first term into <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 = a 1 +d(n–1).
  3. Substitute the last term 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> n and solve for <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>
Finding the Number of Terms in a Finite Arithmetic Sequence

Find the number of terms in the finite arithmetic sequence.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>{</mo><mn>8</mn><mtext>, </mtext><mn>1</mn><mtext>, </mtext><mn>–6</mn><mtext>, </mtext><mn>...</mn><mtext>, </mtext><mn>–41</mn><mo>}</mo></mrow></annotation-xml></semantics></math>

The common difference can be found by subtracting the first term from the second term.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>1</mn><mo>−</mo><mn>8</mn><mo>=</mo><mo>−</mo><mn>7</mn></mrow></annotation-xml></semantics></math>

The common difference is <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>7</mn></mrow></annotation-xml></semantics></math>. Substitute the common difference and the initial term of the sequence into 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> term 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>a</mi></msub></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> n = a 1 + d ( n − 1 ) a n = 8 + − 7 ( n − 1 ) a n = 15 − 7 n

Substitute <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>41</mn></mrow></annotation-xml></semantics></math> 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> n and solve for <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>n</mi></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><mo>−</mo><mn>41</mn><mo>=</mo><mn>15</mn><mo>−</mo><mn>7</mn><mi>n</mi></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math>         8=n

There are eight terms in the sequence.

Find the number of terms in the finite arithmetic sequence.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>{</mo><mn>6</mn><mtext>, </mtext><mn>11</mn><mtext>, </mtext><mn>16</mn><mtext>, </mtext><mn>...</mn><mtext>, </mtext><mn>56</mn><mo>}</mo></mrow></annotation-xml></semantics></math>

There are 11 terms in the sequence.

Solving Application Problems with Arithmetic Sequences

In many application problems, it often makes sense to use an initial term 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> 0 instead 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> 1 . In these problems, we alter the explicit formula slightly to account for the difference in initial terms. We use the following 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> n = a 0 + d n
Solving Application Problems with Arithmetic Sequences

A five-year old child receives an allowance of $1 each week. His parents promise him an annual increase of $2 per week.

  1. Write a formula for the child’s weekly allowance in a given year.
  2. What will the child’s allowance be when he is 16 years old?
  1. The situation can be modeled by an arithmetic sequence with an initial term of 1 and a common difference of 2.

    Let <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>A</mi></annotation-xml></semantics></math> be the amount of the allowance 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> be the number of years after age 5. Using the altered explicit formula for an arithmetic sequence we get:

    <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> n = 1 + 2 n
  2. We can find the number of years since age 5 by subtracting.

    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>16</mn></mrow></annotation-xml></semantics></math> − 5 = 11

    We are looking for the child’s allowance after 11 years. Substitute 11 into the formula to find the child’s allowance at age 16.

    <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> 11 = 1 + 2 ( 11 ) = 23

    The child’s allowance at age 16 will be $23 per week.

A woman decides to go for a 10-minute run every day this week and plans to increase the time of her daily run by 4 minutes each week. Write a formula for the time of her run after n weeks. How long will her daily run be 8 weeks from today?

The formula is <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>T</mi></msub></mrow></annotation-xml></semantics></math> n =10+4n, and it will take her 42 minutes.

Access this online resource for additional instruction and practice with arithmetic sequences.

Key Equations

recursive formula for nth term of an arithmetic sequence <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 = a n−1 +d n≥2
explicit formula for nth term of an arithmetic sequence <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr><mtd><msub><mi>a</mi></msub></mtd></mtr></mtable></annotation-xml></semantics></math> n = a 1 +d(n−1)

Key Concepts

  • An arithmetic sequence is a sequence where the difference between any two consecutive terms is a constant.
  • The constant between two consecutive terms is called the common difference.
  • The common difference is the number added to any one term of an arithmetic sequence that generates the subsequent term. See [link].
  • The terms of an arithmetic sequence can be found by beginning with the initial term and adding the common difference repeatedly. See [link] and [link].
  • A recursive formula for an arithmetic sequence with common difference <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>d</mi></annotation-xml></semantics></math> is given by <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 = a n−1 +d,n≥2. See [link].
  • As with any recursive formula, the initial term of the sequence must be given.
  • An explicit formula for an arithmetic sequence with common difference <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>d</mi></annotation-xml></semantics></math> is given by <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 = a 1 +d(n−1). See [link].
  • An explicit formula can be used to find the number of terms in a sequence. See [link].
  • In application problems, we sometimes alter the explicit formula slightly to <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 = a 0 +dn. See [link].

Section Exercises

Verbal

What is an arithmetic sequence?

A sequence where each successive term of the sequence increases (or decreases) by a constant value.

How is the common difference of an arithmetic sequence found?

How do we determine whether a sequence is arithmetic?

We find whether the difference between all consecutive terms is the same. This is the same as saying that the sequence has a common difference.

What are the main differences between using a recursive formula and using an explicit formula to describe an arithmetic sequence?

Describe how linear functions and arithmetic sequences are similar. How are they different?

Both arithmetic sequences and linear functions have a constant rate of change. They are different because their domains are not the same; linear functions are defined for all real numbers, and arithmetic sequences are defined for natural numbers or a subset of the natural numbers.

Algebraic

For the following exercises, find the common difference for the arithmetic sequence provided.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>{</mo></mrow></annotation-xml></semantics></math> 5 , 11 , 17 , 23 , 29 , ... }

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>{</mo></mrow></mrow></annotation-xml></semantics></math> 0 , 1 2 , 1 , 3 2 , 2 , ... }

The common difference is <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

For the following exercises, determine whether the sequence is arithmetic. If so find the common difference.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>{</mo></mrow></annotation-xml></semantics></math> 11.4 , 9.3 , 7.2 , 5.1 , 3 , ... }

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>{</mo></mrow></annotation-xml></semantics></math> 4 , 16 , 64 , 256 , 1024 , ... }

The sequence is not arithmetic because <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>16</mn><mo>−</mo><mn>4</mn><mo>≠</mo><mn>64</mn><mo>−</mo><mn>16.</mn></mrow></annotation-xml></semantics></math>

For the following exercises, write the first five terms of the arithmetic sequence given the first term and common difference.

<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 =−25, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>d</mi><mo>=</mo><mn>−9</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>a</mi></msub></mrow></annotation-xml></semantics></math> 1 =0, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>d</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 3

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>0</mn><mo>,</mo><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 2 3 ,  4 3 , 2,  8 3

For the following exercises, write the first five terms of the arithmetic series given two terms.

<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 =17,  a 7 =−31

<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> 13 =−60,  a 33 =−160

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>0</mn></mrow></annotation-xml></semantics></math> , − 5 , − 10 , − 15 , − 20

For the following exercises, find the specified term for the arithmetic sequence given the first term and common difference.

First term is 3, common difference is 4, find the 5th term.

First term is 4, common difference is 5, find the 4th term.

<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> 4 =19

First term is 5, common difference is 6, find the 8th term.

First term is 6, common difference is 7, find the 6th term.

<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> 6 =41

First term is 7, common difference is 8, find the 7th term.

For the following exercises, find the first term given two terms from an arithmetic sequence.

Find the first term or <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 of an arithmetic sequence if <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> 6 =12 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> 14 =28.

<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 =2

Find the first term or <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 of an arithmetic sequence if <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> 7 =21 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> 15 =42. 

Find the first term or <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 of an arithmetic sequence if <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 =40 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> 23 =115.

<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

Find the first term or <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 of an arithmetic sequence if <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> 9 =54 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> 17 =102.

Find the first term or <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 of an arithmetic sequence if <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> 11 =11 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> 21 =16.

<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 =6

For the following exercises, find the specified term given two terms from an arithmetic sequence.

<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 =33 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 7 =−15. Find<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 4 .

<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> 3 =−17.1 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 10 =−15.7. 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> 21 .

<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> 21 =−13.5

For the following exercises, use the recursive formula to write the first five terms of the arithmetic sequence.

<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 =39;  a n = a n−1 −3

<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 =−19;  a n = a n−1 −1.4

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>19</mn><mo>,</mo><mo>−</mo><mn>20.4</mn><mo>,</mo><mo>−</mo><mn>21.8</mn><mo>,</mo><mo>−</mo><mn>23.2</mn><mo>,</mo><mo>−</mo><mn>24.6</mn></mrow></annotation-xml></semantics></math>

For the following exercises, write a recursive formula for each arithmetic sequence.

<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 ={ 40,60,80,... }

<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 ={17,26,35,...}

<math display="inline" 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> 1 =17;  a n = a n−1 +9 n≥2

<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 ={−1,2,5,...}

<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 ={12,17,22,...}

<math display="inline" 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> 1 =12;  a n = a n−1 +5 n≥2

<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 ={−15,−7,1,...}

<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> n ={8.9,10.3,11.7,...}

<math display="inline" 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> 1 =8.9;  a n = a n−1 +1.4 n≥2

<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 ={−0.52,−1.02,−1.52,...}

<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 ={ 1 5 , 9 20 , 7 10 ,... }

<math display="inline" 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> 1 = 1 5 ;  a n = a n−1 + 1 4 n≥2

<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 ={ − 1 2 ,− 5 4 ,−2,... }

<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 ={ 1 6 ,− 11 12 ,−2,... }

<math display="inline" 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><mrow/><mn>1</mn></msub></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> = 1 6 ;  a n = a n−1 − 13 12 n≥2

For the following exercises, write a recursive formula for the given arithmetic sequence, and then find the specified term.

<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 ={7, 4, 1, ...}; Find the 17th term.

<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 ={4, 11, 18, ...}; Find the 14th term.

<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;  a n = a n−1 +7;  a 14 =95

<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, 6, 10, ...}; Find the 12th term.

For the following exercises, use the explicit formula to write the first five terms of the arithmetic sequence.

<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 =24−4n

First five terms: <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>20</mn><mo>,</mo><mn>16</mn><mo>,</mo><mn>12</mn><mo>,</mo><mn>8</mn><mo>,</mo><mn>4.</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>a</mi></msub></mrow></annotation-xml></semantics></math> n = 1 2 n− 1 2

For the following exercises, write an explicit formula for each arithmetic sequence.

<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 ={3,5,7,...}

<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 =1+2n

<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 ={32,24,16,...}

<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 ={−5, 95, 195, ...}

<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 =−105+100n

<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 ={−17, −217, −417,...}

<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 ={1.8, 3.6, 5.4, ...}

<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 =1.8n

<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 ={−18.1,−16.2,−14.3,...}

<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 ={15.8,18.5,21.2,...}

<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 =13.1+2.7n

<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 ={ 1 3 ,− 4 3 ,−3, ... }

<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 ={ 0, 1 3 , 2 3 ,... }

<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 = 1 3 n− 1 3

<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 ={ −5,− 10 3 ,− 5 3 ,… }

For the following exercises, find the number of terms in the given finite arithmetic sequence.

<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 ={3,−4,−11, ...,−60}

There are 10 terms in the sequence.

<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 ={1.2,1.4,1.6,...,3.8}

<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 ={ 1 2 ,2, 7 2 ,...,8 }

There are 6 terms in the sequence.

Graphical

For the following exercises, determine whether the graph shown represents an arithmetic sequence.

Graph of a scattered plot with labeled points: (1, -4), (2, -2), (3, 0), (4, 2), and (5, 4). The x-axis is labeled n and the y-axis is labeled a_n.
Graph of a scattered plot with labeled points: (1, 1.5), (2, 2.25), (3, 3.375), (4, 5.0625), and (5, 7.5938). The x-axis is labeled n and the y-axis is labeled a_n.

The graph does not represent an arithmetic sequence.

For the following exercises, use the information provided to graph the first 5 terms of the arithmetic sequence.

<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 =0,d=4

<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 =9; a n = a n−1 −10

Graph of a scattered plot with labeled points: (1, 9), (2, -1), (3, -11), (4, -21), and (5, -31). The x-axis is labeled n and the y-axis is labeled a_n.

<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 =−12+5n

Technology

For the following exercises, follow the steps to work with the arithmetic sequence <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 =3n−2 using a graphing calculator:

  • Press [MODE]
    • Select SEQ in the fourth line
    • Select DOT in the fifth line
    • Press [ENTER]
  • Press [Y=]
    • <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>n</mi><mtext>Min</mtext><mtext> </mtext></mrow></annotation-xml></semantics></math>is the first counting number for the sequence. Set <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>n</mi><mtext>Min</mtext><mo>=</mo><mn>1</mn></mrow></annotation-xml></semantics></math>
    • <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>is the pattern for the sequence. Set <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>u</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mn>3</mn><mi>n</mi><mo>−</mo><mn>2</mn></mrow></annotation-xml></semantics></math>
    • <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>n</mi><mtext>Min)</mtext><mtext> </mtext></mrow></annotation-xml></semantics></math>is the first number in the sequence. Set <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>u</mi><mo stretchy="false">(</mo><mi>n</mi><mtext>Min)</mtext><mo>=</mo><mn>1</mn></mrow></annotation-xml></semantics></math>
  • Press [2ND] then [WINDOW] to go to TBLSET
    • Set<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>TblStart</mtext><mo>=</mo><mn>1</mn></mrow></annotation-xml></semantics></math>
    • Set<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>Δ</mi><mtext>Tbl</mtext><mo>=</mo><mn>1</mn></mrow></annotation-xml></semantics></math>
    • Set Indpnt: Auto and Depend: Auto
  • Press [2ND] then [GRAPH] to go to the TABLE

What are the first seven terms shown in the column with the heading <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mtext>?</mtext></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>1</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>7</mn><mo>,</mo><mn>10</mn><mo>,</mo><mn>13</mn><mo>,</mo><mn>16</mn><mo>,</mo><mn>19</mn></mrow></annotation-xml></semantics></math>

Use the scroll-down arrow to scroll 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>50.</mn></mrow></annotation-xml></semantics></math> What value is given for <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mtext>?</mtext></mrow></annotation-xml></semantics></math>

Press [WINDOW]. Set<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>n</mi><mtext>Min</mtext><mo>=</mo><mn>1</mn><mo>,</mo><mi>n</mi><mtext>Max</mtext><mo>=</mo><mn>5</mn><mo>,</mo><mi>x</mi><mtext>Min</mtext><mo>=</mo><mn>0</mn><mo>,</mo><mi>x</mi><mtext>Max</mtext><mo>=</mo><mn>6</mn><mo>,</mo><mi>y</mi><mtext>Min</mtext><mo>=</mo><mo>−</mo><mn>1</mn><mo>,</mo><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>y</mi><mtext>Max</mtext><mo>=</mo><mn>14.</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>Then press [GRAPH]. Graph the sequence as it appears on the graphing calculator.

Graph of a scattered plot with labeled points: (1, 1), (2, 4), (3, 7), (4, 10), and (5, 13). The x-axis is labeled n and the y-axis is labeled a_n.

For the following exercises, follow the steps given above to work with the arithmetic sequence <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 = 1 2 n+5 using a graphing calculator.

What are the first seven terms shown in the column with the heading<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>u</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>in the TABLE feature?

Graph the sequence as it appears on the graphing calculator. Be sure to adjust the WINDOW settings as needed.

Graph of a scattered plot with labeled points: (1, 5.5), (2, 6), (3, 6.5), (4, 7), and (5, 7.5). The x-axis is labeled n and the y-axis is labeled a_n.

Extensions

Give two examples of arithmetic sequences whose 4th terms are <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>9.</mn></mrow></annotation-xml></semantics></math>

Give two examples of arithmetic sequences whose 10th terms are <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>206.</mn></mrow></annotation-xml></semantics></math>

Answers will vary. Examples: <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 =20.6nand<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+20.4n.

Find the 5th term of the arithmetic sequence <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>{</mo><mn>9</mn><mi>b</mi><mo>,</mo><mn>5</mn><mi>b</mi><mo>,</mo><mi>b</mi><mo>,</mo><mo>…</mo><mo>}</mo><mo>.</mo></mrow></annotation-xml></semantics></math>

Find the 11th term of the arithmetic sequence <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>{</mo><mn>3</mn><mi>a</mi><mo>−</mo><mn>2</mn><mi>b</mi><mo>,</mo><mi>a</mi><mo>+</mo><mn>2</mn><mi>b</mi><mo>,</mo><mo>−</mo><mi>a</mi><mo>+</mo><mn>6</mn><mi>b</mi><mo>…</mo><mo>}</mo><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><msub><mi>a</mi></msub></mrow></annotation-xml></semantics></math> 11 =−17a+38b

At which term does the sequence <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>{</mo><mn>5.4</mn><mo>,</mo><mn>14.5</mn><mo>,</mo><mn>23.6</mn><mo>,</mo><mn>...</mn><mo>}</mo></mrow></annotation-xml></semantics></math> exceed 151?

At which term does the sequence <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>{</mo></mrow></mrow></annotation-xml></semantics></math> 17 3 , 31 6 , 14 3 ,... } begin to have negative values?

The sequence begins to have negative values at the 13th term, <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> 13 =− 1 3

For which terms does the finite arithmetic sequence <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>{</mo></mrow></mrow></annotation-xml></semantics></math> 5 2 , 19 8 , 9 4 ,..., 1 8 } have integer values?

Write an arithmetic sequence using a recursive formula. Show the first 4 terms, and then find the 31st term.

Answers will vary. Check to see that the sequence is arithmetic. Example: Recursive formula: <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 =3, a n = a n−1 −3. First 4 terms: <math display="inline" 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><mn>3</mn><mo>,</mo><mn>0</mn><mo>,</mo><mo>−</mo><mn>3</mn><mo>,</mo><mo>−</mo><mn>6</mn></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> a 31 =−87

Write an arithmetic sequence using an explicit formula. Show the first 4 terms, and then find the 28th term.

Glossary

arithmetic sequence
a sequence in which the difference between any two consecutive terms is a constant
common difference
the difference between any two consecutive terms in an arithmetic sequence