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9.1: Introduction to Sequences and Series

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Skills to Develop

• Find any element of a sequence given a formula for its general term.
• Use sigma notation and expand corresponding series.
• Distinguish between a sequence and a series.
• Calculate the $$n$$th partial sum of sequence.

Sequences

A sequence1 is a function whose domain is a set of consecutive natural numbers beginning with $$1$$. For example, the following equation with domain $$\{1,2,3, \dots\}$$ defines an infinite sequence2:

$$a(n)=5 n-3$$ or $$a_{n}=5 n-3$$

The elements in the range of this function are called terms of the sequence. It is common to define the $$n$$th term, or the general term of a sequence3, using the subscripted notation $$a_{n}$$, which reads “$$a$$ sub $$n$$.” Terms can be found using substitution as follows:

\begin{aligned}\color{Cerulean} { General\: term: } \quad &\color{black}{a_{n}=5 n-3} \\ \color{Cerulean} { First\: term (n=1) :}\quad & \color{black}{a_{1}=}5(\color{Cerulean}{1}\color{black}{)}-3=2 \\ \color{Cerulean}{Second\:term(n=2):} \quad& \color{black}{a_{2}=5}(\color{Cerulean}{2}\color{black}{)}-3=7 \\ \color{Cerulean}{Third\:term(n=3):} \quad& \color{black}{a_{3}=5}(\color{Cerulean}{3}\color{black}{)}-3=12 \\\color{Cerulean}{Fourth\:term(n=4):} \quad& \color{black}{a_{4}=5}(\color{Cerulean}{4}\color{black}{)}-3=17 \\ \color{Cerulean}{Fifth\:term(n=5):} \quad& \color{black}{a_{5}=5}(\color{Cerulean}{5}\color{black}{)}-3=22 \\ \vdots\end{aligned}

This produces an ordered list,

$$2,7,12,17,22, \ldots$$

The ellipsis $$(…)$$ indicates that this sequence continues forever. Unlike a set, order matters. If the domain of a sequence consists of natural numbers that end, such as $$\{1,2,3, \ldots, k\}$$, then it is called a finite sequence4.

Example $$\PageIndex{1}$$:

Given the general term of a sequence, find the first $$5$$ terms as well as the $$100^{th}$$ term: $$a_{n}=\frac{n(n-1)}{2}$$.

Solution:

To find the first $$5$$ terms, substitute $$1, 2, 3, 4$$, and $$5$$ for $$n$$ and then simplify.

$$\begin{array}{l}{a_{1}=\frac{\color{Cerulean}{1}\color{black}{(}\color{Cerulean}{1}\color{black}{-}1)}{2}=\frac{1(0)}{2}=\frac{0}{2}=0} \\ {a_{2}=\frac{\color{Cerulean}{2}\color{black}{(}\color{Cerulean}{2}\color{black}{-}1)}{2}=\frac{2(1)}{2}=\frac{2}{2}=1} \\ {a_{3}=\frac{\color{Cerulean}{3}\color{black}{(}\color{Cerulean}{3}\color{black}{-}1)}{2}=\frac{3(2)}{2}=\frac{6}{2}=3} \\ {a_{4}=\frac{\color{Cerulean}{4}\color{black}{(}\color{Cerulean}{4}\color{black}{-}1)}{2}=\frac{4(3)}{2}=\frac{12}{2}=6} \\ {a_{5}=\frac{\color{Cerulean}{5}\color{black}{(}\color{Cerulean}{5}\color{black}{-}1)}{2}=\frac{5(4)}{2}=\frac{20}{2}=10}\end{array}$$

Use $$n = 100$$ to determine the $$100^{th}$$ term in the sequence.

$$a_{100}=\frac{100(100-1)}{2}=\frac{100(99)}{2}=\frac{9,900}{2}=4,950$$

First five terms: $$0, 1, 3, 6, 10$$; $$a_{100} = 4,950$$

Sometimes the general term of a sequence will alternate in sign and have a variable other than $$n$$.

Example $$\PageIndex{2}$$:

Find the first $$5$$ terms of the sequence: $$a_{n}=(-1)^{n} x^{n+1}$$.

Solution:

Here we take care to replace $$n$$ with the first $$5$$ natural numbers and not $$x$$.

$$\begin{array}{l}{a_{1}=(-1)\color{Cerulean}{^{1}}\color{black}{ x}^{\color{Cerulean}{1}\color{black}{+}1}=-x^{2}} \\ {a_{2}=(-1)^{\color{Cerulean}{2}}\color{black}{ x}^{\color{Cerulean}{2}\color{black}{+}1}=x^{3}} \\ {a_{3}=(-1)^{\color{Cerulean}{3}} \color{black}{x}^{\color{Cerulean}{3}\color{black}{+}1}=-x^{4}} \\ {a_{4}=(-1)^{\color{Cerulean}{4}}\color{black}{ x}^{\color{Cerulean}{4}\color{black}{+}1}=x^{5}} \\ {a_{5}=(-1)^{\color{Cerulean}{5}}\color{black}{ x}^{\color{Cerulean}{5}\color{black}{+}1}=-x^{6}}\end{array}$$

$$-x^{2}, x^{3},-x^{4}, x^{5},-x^{6}$$

Exercise $$\PageIndex{1}$$

Find the first $$5$$ terms of the sequence: $$a_{n}=(-1)^{n+1} 2^{n}$$.

$$2, −4, 8, −16, 32.$$

One interesting example is the Fibonacci sequence. The first two numbers in the Fibonacci sequence are $$1$$, and each successive term is the sum of the previous two. Therefore, the general term is expressed in terms of the previous two as follows:

$$F_{n}=F_{n-2}+F_{n-1}$$

Here $$F_{1} = 1, F_{2} = 1$$, and $$n > 2$$. A formula that describes a sequence in terms of its previous terms is called a recurrence relation5.

Example $$\PageIndex{3}$$:

Find the first $$7$$ Fibonacci numbers.

Solution:

Given that $$F_{1} = 1$$ and $$F_{2} = 1$$, use the recurrence relation $$F_{n}=F_{n-2}+F_{n-1}$$ where $$n$$ is an integer starting with $$n = 3$$ to find the next $$5$$ terms:

$$\begin{array}{l}{F_{3}=F_{\color{Cerulean}{3}\color{black}{-}2}+F_{\color{Cerulean}{3}\color{black}{-}1}=F_{1}+F_{2}=1+1=2} \\ {F_{4}=F_{\color{Cerulean}{4}\color{black}{-}2}+F_{\color{Cerulean}{4}\color{black}{-}1}=F_{2}+F_{3}=1+2=3} \\ {F_{5}=F_{\color{Cerulean}{5}\color{black}{-}2}+F_{\color{Cerulean}{5}\color{black}{-}1}=F_{3}+F_{4}=2+3=5} \\ {F_{6}=F_{\color{Cerulean}{6}\color{black}{-}2}+F_{\color{Cerulean}{6}\color{black}{-}1}=F_{4}+F_{5}=3+5=8} \\ {F_{7}=F_{\color{Cerulean}{7}\color{black}{-}2}+F_{\color{Cerulean}{7}\color{black}{-}1}=F_{5}+F_{6}=5+8=13}\end{array}$$

$$1,1,2,3,5,8,13$$ Figure 9.1.1: Leonardo Fibonacci (1170–1250)

Fibonacci numbers appear in applications ranging from art to computer science and biology. The beauty of this sequence can be visualized by constructing a Fibonacci spiral. Consider a tiling of squares where each side has a length that matches each Fibonacci number: Figure 9.1.2

Connecting the opposite corners of the squares with an arc produces a special spiral shape. Figure 9.1.3

This shape is called the Fibonacci spiral and approximates many spiral shapes found in nature.

Series

A series6 is the sum of the terms of a sequence. The sum of the terms of an infinite sequence results in an infinite series7, denoted $$S_{∞}$$. The sum of the first $$n$$ terms in a sequence is called a partial sum8, denoted $$S_{n}$$. For example, given the sequence of positive odd integers $$1, 3, 5,…$$ we can write:

$$\begin{array}{l}{S_{\infty}=1+3+5+7+9+\dots \quad\color{Cerulean} { Infinite\: series }} \\ {S_{5}=1+3+5+7+9=25 \quad\:\:\color{Cerulean} { 5th\: partial\: sum }}\end{array}$$

Example $$\PageIndex{4}$$:

Determine the $$3^{rd}$$ and $$5^{th}$$ partial sums of the sequence: $$3,−6, 12,−24, 48,…$$

Solution:

$$S_{3}=3+(-6)+12=9$$
$$S_{5}=3+(-6)+12+(-24)+48=33$$

$$S_{3}=9 ; S_{5}=33$$

If the general term is known, then we can express a series using sigma9 (or summation10) notation:

\begin{aligned}S_{\infty}&=\sum_{n=1}^{\infty} n^{2}=1^{2}+2^{2}+3^{2}+\ldots & \color{Cerulean}{Infinite\:series} \\ S_{3}&=\sum_{n=1}^{3} n^{2}=1^{2}+2^{2}+3^{2} & \color{Cerulean}{3rd\:partial\:sum}\end{aligned}

The symbol $$\Sigma$$ (upper case Greek letter sigma) is used to indicate a series. The expressions above and below indicate the range of the index of summation11, in this case represented by $$n$$. The lower number indicates the starting integer and the upper value indicates the ending integer. The $$n$$th partial sum $$S_{n}$$ can be expressed using sigma notation as follows:

$$S_{n}=\sum_{k=1}^{n} a_{k}=a_{1}+a_{2}+\cdots+a_{n}$$

This is read, “the sum of $$a_{k}$$ as $$k$$ goes from $$1$$ to $$n$$.” Replace $$n$$ with $$∞$$ to indicate an infinite sum.

Example $$\PageIndex{5}$$:

Evaluate: $$\sum_{k=1}^{5}(-3)^{n-1}$$.

\begin{aligned} \sum_{k=1}^{5}(-3)^{k-1} &=(-3)^{\color{Cerulean}{1}\color{black}{-}1}+(-3)^{\color{Cerulean}{2}\color{black}{-}1}+(-3)^{\color{Cerulean}{3}\color{black}{-}1}+(-3)^{\color{Cerulean}{4}\color{black}{-}1}+(-3)^{\color{Cerulean}{5}\color{black}{-}1} \\ &=(-3)^{0}+(-3)^{1}+(-3)^{2}+(-3)^{3}+(-3)^{4} \\ &=1-3+9-27+81 \\ &=61 \end{aligned}

$$61$$

When working with sigma notation, the index does not always start at $$1$$.

Example $$\PageIndex{1}$$:

Evaluate: $$\sum_{k=2}^{5}(-1)^{k}(3 k)$$

Solution:

Here the index is expressed using the variable $$k$$, which ranges from $$2$$ to $$5$$.

Exercise $$\PageIndex{2}$$

Evaluate: $$\sum_{n=1}^{5}(15-9 n)$$.

$$-60$$

Infinity is used as the upper bound of a sum to indicate an infinite series.

Example $$\PageIndex{7}$$:

Write in expanded form: $$\sum_{i=0}^{\infty} \frac{n}{n+1}$$.

Solution:

In this case we begin with $$n = 0$$ and add three dots to indicate that this series continues forever.

\begin{aligned} \sum_{n=0}^{\infty} \frac{n}{n+1} &=\frac{\color{Cerulean}{0}}{\color{Cerulean}{0}\color{black}{+}1}+\frac{\color{Cerulean}{1}}{\color{Cerulean}{1}\color{black}{+}1}+\frac{\color{Cerulean}{2}}{\color{Cerulean}{2}\color{black}{+}1}+\frac{\color{Cerulean}{3}}{\color{Cerulean}{3}\color{black}{+}1}+\cdots \\ &=\frac{0}{1}+\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\cdots \\ &=0+\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\cdots \end{aligned}

$$0+\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\cdots$$

When expanding a series, take care to replace only the variable indicated by the index.

Example $$\PageIndex{8}$$:

Write in expanded form: $$\sum_{i=1}^{\infty}(-1)^{i-1} x^{2 i}$$.

Solution:

\begin{aligned} \sum_{i=1}^{\infty}(-1)^{i-1} x^{2 i} &=(-1)^{\color{Cerulean}{1}\color{black}{-}1} x^{2(\color{Cerulean}{1}\color{black}{)}}+(-1)^{\color{Cerulean}{2}\color{black}{-}1} x^{2(\color{Cerulean}{2}\color{black}{)}}+(-1)^{\color{Cerulean}{3}\color{black}{-}1} x^{2(\color{Cerulean}{3}\color{black}{)}}+\cdots \\ &=(-1)^{0} x^{2(1)}+(-1)^{1} x^{2(2)}+(-1)^{2} x^{2(3)}+\cdots \\ &=x^{2}-x^{4}+x^{6}-\cdots \end{aligned}

$$x^{2}-x^{4}+x^{6}-\cdots$$

Key Takeaways

• A sequence is a function whose domain consists of a set of natural numbers beginning with $$1$$. In addition, a sequence can be thought of as an ordered list.
• Formulas are often used to describe the $$n$$th term, or general term, of a sequence using the subscripted notation $$a_{n}$$.
• A series is the sum of the terms in a sequence. The sum of the first $$n$$ terms is called the $$n$$th partial sum and is denoted $$S_{n}$$.
• Use sigma notation to denote summations in a compact manner. The nth partial sum, using sigma notation, can be written $$S_{n}=\sum_{k=1}^{n} a_{k}$$. The symbol $$\Sigma$$ denotes a summation where the expression below indicates that the index $$k$$ starts at $$1$$ and iterates through the natural numbers ending with the value $$n$$ above.

Exercise $$\PageIndex{3}$$

Find the first $$5$$ terms of the sequence as well as the $$30^{th}$$ term.

1. $$a_{n}=2 n$$
2. $$a_{n}=2 n+1$$
3. $$a_{n}=\frac{n^{2}-1}{2}$$
4. $$a_{n}=\frac{n}{2 n-1}$$
5. $$a_{n}=(-1)^{n}(n+1)^{2}$$
6. $$a_{n}=(-1)^{n+1} n^{2}$$
7. $$a_{n}=3^{n-1}$$
8. $$a_{n}=2^{n-2}$$
9. $$a_{n}=\left(\frac{1}{2}\right)^{n}$$
10. $$a_{n}=\left(-\frac{1}{3}\right)^{n}$$
11. $$a_{n}=\frac{(-1)^{n-1}}{3 n-1}$$
12. $$a_{n}=\frac{2(-1)^{n}}{n+5}$$
13. $$a_{n}=1+\frac{1}{n}$$
14. $$a_{n}=\frac{n^{2}+1}{n}$$

1. $$2,4,6,8,10 ; a_{30}=60$$

3. $$0, \frac{3}{2}, 4, \frac{15}{2}, 12 ; a_{30}=\frac{899}{2}$$

5. $$-4,9,-16,25,-36 ; a_{30}=961$$

7. $$1,3,9,27,81 ; a_{30}=3^{29}$$

9. $$\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32} ; a_{30}=\frac{1}{2^{30}}$$

11. $$\frac{1}{2},-\frac{1}{5}, \frac{1}{8},-\frac{1}{11}, \frac{1}{14} ; a_{30}=-\frac{1}{89}$$

13. $$2, \frac{3}{2}, \frac{4}{3}, \frac{5}{4}, \frac{6}{5} ; a_{30}=\frac{31}{30}$$

Exercise $$\PageIndex{4}$$

Find the first $$5$$ terms of the sequence.

1. $$a_{n}=2 x^{2 n-1}$$
2. $$a_{n}=(2 x)^{n-1}$$
3. $$a_{n}=\frac{x^{n}}{n+4}$$
4. $$a_{n}=\frac{x^{2 n}}{x-2}$$
5. $$a_{n}=\frac{n x^{2 n}}{n+1}$$
6. $$a_{n}=\frac{(n+1) x^{n}}{n^{2}}$$
7. $$a_{n}=(-1)^{n} x^{3 n}$$
8. $$a_{n}=(-1)^{n-1} x^{n+1}$$

1. $$2 x, 2 x^{3}, 2 x^{5}, 2 x^{7}, 2 x^{9}$$

3. $$\frac{x}{5}, \frac{x^{2}}{6}, \frac{x^{3}}{7}, \frac{x^{4}}{8}, \frac{x^{5}}{9}$$

5. $$\frac{x^{2}}{2}, \frac{2 x^{4}}{3}, \frac{3 x^{6}}{4}, \frac{4 x^{8}}{5}, \frac{5 x^{10}}{6}$$

7. $$-x^{3}, x^{6},-x^{9}, x^{12},-x^{15}$$

Exercise $$\PageIndex{5}$$

Find the first 5 terms of the sequence defined by the given recurrence relation.

1. $$a_{n}=a_{n-1}+5$$ where $$a_{1}=3$$
2. $$a_{n}=a_{n-1}-3$$ where $$a_{1}=4$$
3. $$a_{n}=3 a_{n-1}$$ where $$a_{1}=-2$$
4. $$a_{n}=-2 a_{n-1}$$ where $$a_{1}=-1$$
5. $$a_{n}=n a_{n-1}$$ where $$a_{1}=1$$
6. $$a_{n}=(n-1) a_{n-1}$$ where $$a_{1}=1$$
7. $$a_{n}=2 a_{n-1}-1$$ where $$a_{1}=0$$
8. $$a_{n}=3 a_{n-1}+1$$ where $$a_{1}=-1$$
9. $$a_{n}=a_{n-2}+2 a_{n-1}$$ where $$a_{1}=-1$$ and $$a_{2}=0$$
10. $$a_{n}=3 a_{n-1}-a_{n-2}$$ where $$a_{1}=0$$ and $$a_{2}=2$$
11. $$a_{n}=a_{n-1}-a_{n-2}$$ where $$a_{1}=1$$ and $$a_{2}=3$$
12. $$a_{n}=a_{n-2}+a_{n-1}+2$$ where $$a_{1}=-4$$ and $$a_{2}=-1$$

1. $$3, 8, 13, 18, 23$$

3. $$−2, −6, −18, −54, −162$$

5. $$1, 2, 6, 24, 120$$

7. $$0, −1, −3, −7, −15$$

9. $$−1, 0, −1, −2, −5$$

11. $$1, 3, 2, −1, −3$$

Exercise $$\PageIndex{6}$$

Find the indicated term.

1. $$a_{n}=2-7 n ; a_{12}$$
2. $$a_{n}=3 n-8 ; a_{20}$$
3. $$a_{n}=-4(5)^{n-4} ; a_{7}$$
4. $$a_{n}=6\left(\frac{1}{3}\right)^{n-6} ; a_{9}$$
5. $$a_{n}=1+\frac{1}{n}; a_{10}$$
6. $$a_{n}=(n+1) 5^{n-3} ; a_{5}$$
7. $$a_{n}=(-1)^{n} 2^{2 n-3} ; a_{4}$$
8. $$a_{n}=n(n-1)(n-2) ; a_{6}$$
9. An investment of $$$4,500$$ is made in an account earning $$2$$% interest compounded quarterly. The balance in the account after $$n$$ quarters is given by $$a_{n}=4500\left(1+\frac{0.02}{4}\right)^{n}$$. Find the amount in the account after each quarter for the first two years. Round to the nearest cent. 10. The value of a new car after $$n$$ years is given by the formula $$a_{n}=18,000\left(\frac{3}{4}\right)^{n}$$. Find and interpret $$a_{7}$$. Round to the nearest whole dollar. 11. The number of comparisons a computer algorithm makes to sort n names in a list is given by the formula $$a_{n}=n \log _{2} n$$. Determine the number of comparisons it takes this algorithm to sort $$2 × 10^{6}$$ (2 million) names. 12. The number of comparisons a computer algorithm makes to search $$n$$ names in a list is given by the formula $$a_{n}=n^{2}$$ Determine the number of comparisons it takes this algorithm to search $$2 × 10^{6}$$ (2 million) names. Answer 1. $$-82$$ 3. $$-500$$ 5. $$\frac{11}{10}$$ 7. $$32$$ 9. Year 1: QI:$$$4,522.50$$; QII: $$$4,545.11$$; QIII:$$$4,567.84$$; QIV: $$$4,590.68$$; Year 2: QI:$$$4,613.63$$; QII: $$$4,636.70$$; QIII:$$$4,659.88$$; QIV: \$$$4,683.18$$

11. Approximately $$4 \times 10^{7}$$ comparisons

Exercise $$\PageIndex{7}$$

Find the indicated partial sum.

1. $$3,5,9,17,33, \ldots ; S_{4}$$
2. $$-5,7,-29,79,-245, \ldots ; S_{4}$$
3. $$4,1,-4,-11,-20, \ldots ; S_{5}$$
4. $$0,2,6,12,20, \ldots ; S_{3}$$
5. $$a_{n}=2-7 n ; S_{5}$$
6. $$a_{n}=3 n-8 ; S_{5}$$
7. $$a_{n}=-4(5)^{n-4} ; S_{3}$$
8. $$a_{n}=6\left(\frac{1}{3}\right)^{n-6} ; S_{3}$$
9. $$a_{n}=1+\frac{1}{n}; S_{4}$$
10. $$a_{n}=(n+1) 5^{n-3} ; S_{3}$$
11. $$a_{n}=(-1)^{n} 2^{2 n-3} ; S_{5}$$
12. $$a_{n}=n(n-1)(n-2) ; S_{4}$$

1. $$34$$

3. $$-30$$

5. $$-95$$

7. $$-\frac{124}{125}$$

9. $$\frac{73}{12}$$

11. $$-\frac{205}{2}$$

Exercise $$\PageIndex{8}$$

Evaluate.

1. $$\sum_{k=1}^{5} 3 k$$
2. $$\sum_{k=1}^{6} 2 k$$
3. $$\sum_{i=2}^{6} i^{2}$$
4. $$\sum_{i=0}^{4}(i+1)^{2}$$
5. $$\sum_{n=1}^{5}(-1)^{n+1} 2^{n}$$
6. $$\sum_{n=5}^{10}(-1)^{n} n^{2}$$
7. $$\sum_{k=-2}^{2}\left(\frac{1}{2}\right)^{k}$$
8. $$\sum_{k=-4}^{0}\left(\frac{1}{3}\right)^{k}$$
9. $$\sum_{k=0}^{4}(-2)^{k+1}$$
10. $$\sum_{k=-1}^{3}(-3)^{k-1}$$
11. $$\sum_{n=1}^{5} 3$$
12. $$\sum_{n=1}^{7}-5$$
13. $$\sum_{k=-2}^{3} k(k+1)$$
14. $$\sum_{k=-2}^{2}(k-2)(k+2)$$

1. $$45$$

3. $$90$$

5. $$22$$

7. $$\frac{31}{4}$$

9. $$−22$$

11. $$15$$

13. $$22$$

Exercise $$\PageIndex{9}$$

Write in expanded form.

1. $$\sum_{n=1}^{\infty} \frac{n-1}{n}$$
2. $$\sum_{n=1}^{\infty} \frac{n}{2 n-1}$$
3. $$\sum_{n=1}^{\infty}\left(-\frac{1}{2}\right)^{n-1}$$
4. $$\sum_{n=0}^{\infty}\left(-\frac{2}{3}\right)^{n+1}$$
5. $$\sum_{n=1}^{\infty} 3\left(\frac{1}{5}\right)^{n}$$
6. $$\sum_{n=0}^{\infty} 2\left(\frac{1}{3}\right)^{n}$$
7. $$\sum_{k=0}^{\infty}(-1)^{k} x^{k+1}$$
8. $$\sum_{k=1}^{\infty}(-1)^{k+1} x^{k-1}$$
9. $$\sum_{i=0}^{\infty}(-2)^{i+1} x^{i}$$
10. $$\sum_{i=1}^{\infty}(-3)^{i-1} x^{3 i}$$
11. $$\sum_{k=1}^{\infty}(2 k-1) x^{2 k}$$
12. $$\sum_{k=1}^{\infty} \frac{k x^{k-1}}{k+1}$$

1. $$0+\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\cdots$$

3. $$1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots$$

5. $$\frac{3}{5}+\frac{3}{25}+\frac{3}{125}+\frac{3}{625}+\cdots$$

7. $$x-x^{2}+x^{3}-x^{4}+\cdots$$

9. $$-2+4 x-8 x^{2}+16 x^{3}-\dots$$

11. $$x^{2}+3 x^{4}+5 x^{6}+7 x^{8}+\cdots$$

Exercise $$\PageIndex{10}$$

Express the following series using sigma notation.

1. $$x+2 x^{2}+3 x^{3}+4 x^{4}+5 x^{5}$$
2. $$\frac{1}{2} x^{2}+\frac{2}{3} x^{3}+\frac{3}{4} x^{4}+\frac{4}{5} x^{5}+\frac{5}{6} x^{6}$$
3. $$2+2^{2} x+2^{3} x^{2}+2^{4} x^{3}+2^{5} x^{4}$$
4. $$3 x+3^{2} x^{2}+3^{3} x^{3}+3^{4} x^{4}+3^{5} x^{5}$$
5. $$2 x+4 x^{2}+8 x^{3}+\dots+2^{n} x^{n}$$
6. $$x+3 x^{2}+9 x^{3}+\dots+3^{n} x^{n+1}$$
7. $$5+(5+d)+(5+2 d)+\dots+(5+n d)$$
8. $$2+2 r^{1}+2 r^{2}+\dots+2 r^{n-1}$$
9. $$\frac{3}{4}+\frac{3}{8}+\frac{3}{16}+\dots+3\left(\frac{1}{2}\right)^{n}$$
10. $$\frac{8}{3}+\frac{16}{4}+\frac{32}{5}+\dots+\frac{2^{n}}{n}$$
11. A structured settlement yields an amount in dollars each year, represented by $$n$$, according to the formula $$p_{n}=10,000(0.70)^{n-1}$$. What is the total amount gained from the settlement after $$5$$ years?
12. The first row of seating in a small theater consists of $$14$$ seats. Each row thereafter consists of $$2$$ more seats than the previous row. If there are $$7$$ rows, how many total seats are in the theater?

1. $$\sum_{k=1}^{5} k x^{k}$$

3. $$\sum_{k=1}^{5} 2^{k} x^{k-1}$$

5. $$\sum_{k=1}^{n} 2^{k} x^{k}$$

7. $$\sum_{k=0}^{n}(5+k d)$$

9. $$\sum_{k=2}^{n} 3\left(\frac{1}{2}\right)^{k}$$

11. $$\ 27,731$$

Exercise $$\PageIndex{11}$$

1. Research and discuss Fibonacci numbers as they are found in nature.
2. Research and discuss the life and contributions of Leonardo Fibonacci.
3. Explain the difference between a sequence and a series. Provide an example of each.

Footnotes

1A function whose domain is a set of consecutive natural numbers starting with $$1$$.

2A sequence whose domain is the set of natural numbers $$\{1,2,3, \dots\}$$.

3An equation that defines the nth term of a sequence commonly denoted using subscripts $$a_{n}$$.

4A sequence whose domain is $$\{1,2,3, \dots, k\}$$ where $$k$$ is a natural number.

5A formula that uses previous terms of a sequence to describe subsequent terms.

6The sum of the terms of a sequence.

7The sum of the terms of an infinite sequence denoted $$S_{∞}$$.

8The sum of the first n terms in a sequence denoted $$S_{n}$$.

9A sum denoted using the symbol $$\Sigma$$ (upper case Greek letter sigma).

10Used when referring to sigma notation.

11The variable used in sigma notation to indicate the lower and upper bounds of the summation.