$\mathrm{An}$ _____ _____ is a function whose domain is the set of positive integers.

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A sequence is a _____ sequence when the domain of the function consists only of the first $n$ positive integers.

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If you are given one or more of the first few terms of a sequence, and all other terms of the sequence are defined using previous terms, then the sequence is said to be defined _____.

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If $n$ is a positive integer, then $n$ _____ is defined as $n !=1 \cdot 2 \cdot 3 \cdot 4 \cdots(n-1) \cdot n$

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For the sum $$\sum_{i=1}^{n} a_{i}, $$ $i$ is called the _____ of summation, $n$ is the _____ limit of summation, and 1 is the _____ limit of summation.

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Writing the Terms of a Sequence In Exercises

$7-22,$ write the first five terms of the sequence. (Assume

that $n$ begins with $1 .$ )

$$

a_{n}=4 n-7

$$

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Writing the Terms of a Sequence In Exercises

$7-22,$ write the first five terms of the sequence. (Assume

that $n$ begins with $1 .$ )

$$

a_{n}=2-\frac{1}{3^{n}}

$$

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Writing the Terms of a Sequence In Exercises

$7-22,$ write the first five terms of the sequence. (Assume

that $n$ begins with $1 .$ )

$$

a_{n}=(-2)^{n}

$$

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Writing the Terms of a Sequence In Exercises

$7-22,$ write the first five terms of the sequence. (Assume

that $n$ begins with $1 .$ )

$$

a_{n}=\left(\frac{1}{2}\right)^{n}

$$

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Writing the Terms of a Sequence In Exercises

$7-22,$ write the first five terms of the sequence. (Assume

that $n$ begins with $1 .$ )

$$

a_{n}=\frac{n}{n+2}

$$

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Writing the Terms of a Sequence In Exercises

$7-22,$ write the first five terms of the sequence. (Assume

that $n$ begins with $1 .$ )

$$

a_{n}=\frac{6 n}{3 n^{2}-1}

$$

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Writing the Terms of a Sequence In Exercises

$7-22,$ write the first five terms of the sequence. (Assume

that $n$ begins with $1 .$ )

$$

a_{n}=\frac{1+(-1)^{n}}{n}

$$

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Writing the Terms of a Sequence In Exercises

$7-22,$ write the first five terms of the sequence. (Assume

that $n$ begins with $1 .$ )

$$

a_{n}=\frac{(-1)^{n}}{n^{2}}

$$

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Writing the Terms of a Sequence In Exercises

$7-22,$ write the first five terms of the sequence. (Assume

that $n$ begins with $1 .$ )

$$

a_{n}=\frac{2^{n}}{3^{n}}

$$

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Writing the Terms of a Sequence In Exercises

$7-22,$ write the first five terms of the sequence. (Assume

that $n$ begins with $1 .$ )

$$

a_{n}=\frac{1}{n^{3 / 2}}

$$

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Writing the Terms of a Sequence In Exercises

$7-22,$ write the first five terms of the sequence. (Assume

that $n$ begins with $1 .$ )

$$

a_{n}=\frac{2}{3}

$$

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Writing the Terms of a Sequence In Exercises

$7-22,$ write the first five terms of the sequence. (Assume

that $n$ begins with $1 .$ .

$$

a_{n}=1+(-1)^{n}

$$

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Writing the Terms of a Sequence In Exercises

$7-22,$ write the first five terms of the sequence. (Assume

that $n$ begins with $1 .$ )

$$

a_{n}=n(n-1)(n-2)

$$

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Writing the Terms of a Sequence In Exercises

$7-22,$ write the first five terms of the sequence. (Assume

that $n$ begins with $1 .$ )

$$

a_{n}=n\left(n^{2}-6\right)

$$

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Writing the Terms of a Sequence In Exercises

$7-22,$ write the first five terms of the sequence. (Assume

that $n$ begins with $1 .$ )

$$

a_{n}=(-1)^{n}\left(\frac{n}{n+1}\right)

$$

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Writing the Terms of a Sequence In Exercises

$7-22,$ write the first five terms of the sequence. (Assume

that $n$ begins with $1 .$ )

$$

a_{n}=\frac{(-1)^{n+1}}{n^{2}+1}

$$

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Finding a Term of a Sequence In Exercises $23-26$ ,

find the indicated term of the sequence.

$$

\begin{array}{l}{a_{n}=(-1)^{n}(3 n-2)} \\ {a_{25}=}\end{array}

$$

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Finding a Term of a Sequence In Exercises $23-26$

find the indicated term of the sequence.

$$

\begin{array}{l}{a_{n}=(-1)^{n-1}[n(n-1)]} \\ {a_{16}=}\end{array}

$$

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Finding a Term of a Sequence In Exercises $23-26$

find the indicated term of the sequence.

$$

\begin{array}{l}{a_{n}=\frac{4 n}{2 n^{2}-3}} \\ {a_{11}=}\end{array}

$$

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Finding a Term of a Sequence In Exercises $23-26$

find the indicated term of the sequence.

$$

\begin{array}{l}{a_{n}=\frac{4 n^{2}-n+3}{n(n-1)(n+2)}} \\ {a_{13}=}\end{array}

$$

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Graphing the Terms of a Sequence In Exercises

$27-32,$ use a graphing utility to graph the first 10 terms

of the sequence. (Assume that $n$ begins with $1 . )$

$$

a_{n}=\frac{2}{3} n

$$

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Graphing the Terms of a Sequence In Exercises

$27-32,$ use a graphing utility to graph the first 10 terms

of the sequence. (Assume that $n$ begins with $1 .$ .

$$

a_{n}=2-\frac{4}{n}

$$

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Graphing the Terms of a Sequence In Exercises

$27-32,$ use a graphing utility to graph the first 10 terms

of the sequence. (Assume that $n$ begins with $1 . )$

$$

a_{n}=16(-0.5)^{n-1}

$$

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Graphing the Terms of a Sequence In Exercises

$27-32,$ use a graphing utility to graph the first 10 terms

of the sequence. (Assume that $n$ begins with $1 . )$

$$

a_{n}=8(0.75)^{n-1}

$$

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Graphing the Terms of a Sequence In Exercises

$27-32,$ use a graphing utility to graph the first 10 terms

of the sequence. (Assume that $n$ begins with $1 . )$

$$

a_{n}=\frac{2 n}{n+1}

$$

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Graphing the Terms of a Sequence In Exercises

$27-32,$ use a graphing utility to graph the first 10 terms

of the sequence. (Assume that $n$ begins with $1 . )$

$$

a_{n}=\frac{3 n^{2}}{n^{2}+1}

$$

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Matching a Sequence with a Graph In Exercises

$33-36,$ match the sequence with the graph of its first 10

terms. [The graphs are labeled (a), (b), (c), and (d).]

$$

a_{n}=\frac{8}{n+1}

$$

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Matching a Sequence with a Graph In Exercises

$33-36$ , match the sequence with the graph of its first 10

terms. [The graphs are labeled (a), (b), (c), and (d).]

$$

a_{n}=\frac{8 n}{n+1}

$$

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Matching a Sequence with a Graph In Exercises

$33-36$ , match the sequence with the graph of its first 10

terms. [The graphs are labeled (a), ( b), (c), and (d).]

$$

a_{n}=4(0.5)^{n-1}

$$

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Matching a Sequence with a Graph In Exercises

$33-36,$ match the sequence with the graph of its first 10

terms. [The graphs are labeled (a), (b), (c), and (d).]

$$

a_{n}=\frac{4^{n}}{n !}

$$

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Finding the $n$ th Term of a Sequence In Exercises

$37-48,$ write an expression for the apparent $n$ th term

$\left(a_{n}\right)$ of the sequence. (Assume that $n$ begins with $1 . )$

$$

3,7,11,15,19, \ldots

$$

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Finding the $n$ th Term of a Sequence In Exercises

$37-48,$ write an expression for the apparent $n$ th term

$\left(a_{n}\right)$ of the sequence. (Assume that $n$ begins with $1 . )$

$$

0,3,8,15,24, \dots

$$

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Finding the $n$ th Term of a Sequence In Exercises

$37-48,$ write an expression for the apparent $n$ th term

$\left(a_{n}\right)$ of the sequence. (Assume that $n$ begins with $1 . )$

$$

-\frac{2}{3}, \frac{3}{4},-\frac{4}{5}, \frac{5}{6},-\frac{6}{7}, \dots

$$

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Finding the $n$ th Term of a Sequence In Exercises

$37-48,$ write an expression for the apparent $n$ th term

$\left(a_{n}\right)$ of the sequence. (Assume that $n$ begins with $1 . )$

$$

\frac{1}{2},-\frac{1}{4}, \frac{1}{8},-\frac{1}{16}, \ldots

$$

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Finding the $n$ th Term of a Sequence In Exercises

$37-48,$ write an expression for the apparent $n$ th term

$\left(a_{n}\right)$ of the sequence. (Assume that $n$ begins with $1 . )$

$$

\frac{2}{1}, \frac{3}{3}, \frac{4}{5}, \frac{5}{7}, \frac{6}{9}, \ldots

$$

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Finding the $n$ th Term of a Sequence In Exercises

$37-48,$ write an expression for the apparent $n$ th term

$\left(a_{n}\right)$ of the sequence. (Assume that $n$ begins with $1 . )$

$$

\frac{1}{3}, \frac{2}{9}, \frac{4}{27}, \frac{8}{81}, \dots

$$

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Finding the $n$ th Term of a Sequence In Exercises

$37-48,$ write an expression for the apparent $n$ th term

$\left(a_{n}\right)$ of the sequence. (Assume that $n$ begins with $1 . )$

$$

1, \frac{1}{4}, \frac{1}{9}, \frac{1}{16}, \frac{1}{25}, \dots

$$

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Finding the $n$ th Term of a Sequence In Exercises

$37-48,$ write an expression for the apparent $n$ th term

$\left(a_{n}\right)$ of the sequence. (Assume that $n$ begins with $1 . )$

$$

1, \frac{1}{2}, \frac{1}{6}, \frac{1}{24}, \frac{1}{120}, \dots

$$

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Finding the $n$ th Term of a Sequence In Exercises

$37-48,$ write an expression for the apparent $n$ th term

$\left(a_{n}\right)$ of the sequence. (Assume that $n$ begins with $1 . )$

$$

1,-1,1,-1,1, \ldots

$$

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Finding the $n$ th Term of a Sequence In Exercises

$37-48,$ write an expression for the apparent $n$ th term

$\left(a_{n}\right)$ of the sequence. (Assume that $n$ begins with $1 . )$

$$

1,3,1,3,1, \dots

$$

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Finding the $n$ th Term of a Sequence In Exercises

$37-48,$ write an expression for the apparent $n$ th term

$\left(a_{n}\right)$ of the sequence. (Assume that $n$ begins with $1 . )$

$$

1,3, \frac{3^{2}}{2}, \frac{3^{3}}{6}, \frac{3^{4}}{24}, \frac{3^{5}}{120}, \dots

$$

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Finding the $n$ th Term of a Sequence In Exercises

$37-48,$ write an expression for the apparent $n$ th term

$\left(a_{n}\right)$ of the sequence. (Assume that $n$ begins with $1 . )$

$$

1+\frac{1}{2}, 1+\frac{3}{4}, 1+\frac{7}{8}, 1+\frac{15}{16}, 1+\frac{31}{32}, \ldots

$$

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Writing the Terms of a Recursive Sequence In

Exercises $49-52,$ write the first five terms of the sequence

defined recursivelv.

$$

a_{1}=28, \quad a_{k+1}=a_{k}-4

$$

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Writing the Terms of a Recursive Sequence In

Exercises $49-52$ , write the first five terms of the sequence

defined recursively.

$$

a_{1}=3, \quad a_{k+1}=2\left(a_{k}-1\right)

$$

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Writing the Terms of a Recursive Sequence In

Exercises $49-52$ , write the first five terms of the sequence

defined recursively.

$$

a_{0}=1, a_{1}=2, \quad a_{k}=a_{k-2}+\frac{1}{2} a_{k-1}

$$

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Writing the Terms of a Recursive Sequence In

Exercises $49-52,$ write the first five terms of the sequence

defined recursively.

$$

a_{0}=-1, a_{1}=1, \quad a_{k}=a_{k-2}+a_{k-1}

$$

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Writing the $n$ th Term of a Recursive Sequence

In Exercises $53-56$ , write the first five terms of the

sequence defined recursively. Use the pattern to write the

$n$ th term of the sequence as a function of $n .$

$$

a_{1}=6, \quad a_{k+1}=a_{k}+2

$$

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Writing the $n$ th Term of a Recursive Sequence

In Exercises $53-56$ , write the first five terms of the

sequence defined recursively. Use the pattern to write the

$n$ th term of the sequence as a function of $n .$

$$

a_{1}=25, \quad a_{k+1}=a_{k}-5

$$

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Writing the $n$ th Term of a Recursive Sequence

In Exercises $53-56$ , write the first five terms of the

sequence defined recursively. Use the pattern to write the

$n$ th term of the sequence as a function of $n .$

$$

a_{1}=81, \quad a_{k+1}=\frac{1}{3} a_{k}

$$

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Writing the $n$ th Term of a Recursive Sequence

In Exercises $53-56$ , write the first five terms of the

sequence defined recursively. Use the pattern to write the

$n$ th term of the sequence as a function of $n .$

$$

a_{1}=14, \quad a_{k+1}=(-2) a_{k}

$$

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Fibonacci Sequence In Exercises 57 and 58 , use the

Fibonacci sequence. (See Example 5.)

Write the first 12 terms of the Fibonacci sequence $a_{n}$

and the first 10 terms of the sequence given by

$$

b_{n}=\frac{a_{n+1}}{a_{n}}, \quad n \geq 1

$$

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Fibonacci Sequence In Exercises 57 and 58 , use the

Fibonacci sequence. (See Example $5 . )$

Using the definition for $b_{n}$ in Exercise $57,$ show that $b_{n}$

can be defined recursively by

$$

b_{n}=1+\frac{1}{b_{n-1}}

$$

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Writing the Terms of a Sequence Involving

Factorials In Exercises $59-62,$ write the first five terms

of the sequence. (Assume that $n$ begins with $0 . )$

$$

a_{n}=\frac{5}{n !}

$$

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Writing the Terms of a Sequence Involving

Factorials In Exercises $59-62,$ write the first five terms

of the sequence. (Assume that $n$ begins with $0 . )$

$$

a_{n}=\frac{n !}{2 n+1}

$$

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Writing the Terms of a Sequence Involving

Factorials In Exercises $59-62,$ write the first five terms

of the sequence. (Assume that $n$ begins with $0 . )$

$$

a_{n}=\frac{1}{(n+1) !}

$$

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Writing the Terms of a Sequence Involving

Factorials In Exercises $59-62,$ write the first five terms

of the sequence. (Assume that $n$ begins with $0 . )$

$$

a_{n}=\frac{(-1)^{2 n+1}}{(2 n+1) !}

$$

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Simplifying a Factorial Expression In Exercises

$63-66,$ simplify the factorial expression.

$$

\frac{4 !}{6 !}

$$

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Simplifying a Factorial Expression In Exercises

$63-66,$ simplify the factorial expression.

$$

\frac{12 !}{4 ! \cdot 8 !}

$$

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Simplifying a Factorial Expression In Exercises

$63-66,$ simplify the factorial expression.

$$

\frac{(n+1) !}{n !}

$$

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Simplifying a Factorial Expression In Exercises

$63-66,$ simplify the factorial expression.

$$

\frac{(2 n-1) !}{(2 n+1) !}

$$

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Finding a Sum In Exercises $67-74,$ find the sum.

$$

\sum_{i=1}^{5}(2 i+1)

$$

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Finding a Sum In Exercises $67-74,$ find the sum.

$$

\sum_{j=3}^{5} \frac{1}{j^{2}-3}

$$

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Finding a Sum In Exercises $67-74,$ find the sum.

$$

\sum_{i=0}^{4} i^{2}

$$

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Finding a Sum In Exercises $67-74,$ find the sum.

$$

\sum_{k=2}^{5}(k+1)^{2}(k-3)

$$

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Finding a Sum In Exercises $67-74,$ find the sum.

$$

\sum_{i=1}^{4}\left[(i-1)^{2}+(i+1)^{3}\right]

$$

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Finding a Sum In Exercises $67-74,$ find the sum.

$$

\sum_{i=1}^{4} 2^{i}

$$

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Finding a Sum In Exercises $67-74,$ find the sum.

$$

\sum_{j=0}^{4}(-2)^{j}

$$

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Finding a Sum In Exercises $75-78$ , use a graphing

utility to find the sum.

$$

\sum_{n=0}^{5} \frac{1}{2 n+1}

$$

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Finding a Sum In Exercises $75-78$ , use a graphing

utility to find the sum.

$$

\sum_{k=0}^{4} \frac{(-1)^{k}}{k+1}

$$

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Finding a Sum In Exercises $75-78$ , use a graphing

utility to find the sum.

$$

\sum_{k=0}^{4} \frac{(-1)^{k}}{k !}

$$

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Finding a Sum In Exercises $75-78$ , use a graphing

utility to find the sum.

$$

\sum_{n=0}^{25} \frac{1}{4^{n}}

$$

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Using Sigma Notation to Write a Sum In

Exercises $79-88$ , use sigma notation to write the sum.

$$

\frac{1}{3(1)}+\frac{1}{3(2)}+\frac{1}{3(3)}+\dots+\frac{1}{3(9)}

$$

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Using Sigma Notation to Write a Sum In

Exercises $79-88$ , use sigma notation to write the sum.

$$

\frac{5}{1+1}+\frac{5}{1+2}+\frac{5}{1+3}+\dots+\frac{5}{1+15}

$$

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Using Sigma Notation to Write a Sum In

Exercises $79-88$ , use sigma notation to write the sum.

$$

\left[2\left(\frac{1}{8}\right)+3\right]+\left[2\left(\frac{2}{8}\right)+3\right]+\cdots+\left[2\left(\frac{8}{8}\right)+3\right]

$$

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Using Sigma Notation to Write a Sum In

Exercises $79-88$ , use sigma notation to write the sum.

$$

\left[1-\left(\frac{1}{6}\right)^{2}\right]+\left[1-\left(\frac{2}{6}\right)^{2}\right]+\cdots+\left[1-\left(\frac{6}{6}\right)^{2}\right]

$$

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Using Sigma Notation to Write a Sum In

Exercises $79-88$ , use sigma notation to write the sum.

$$

3-9+27-81+243-729

$$

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Using Sigma Notation to Write a Sum In

Exercises $79-88$ , use sigma notation to write the sum.

$$

1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\dots-\frac{1}{128}

$$

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Using Sigma Notation to Write a Sum In

Exercises $79-88$ , use sigma notation to write the sum.

$$

\frac{1}{1^{2}}-\frac{1}{2^{2}}+\frac{1}{3^{2}}-\frac{1}{4^{2}}+\ldots-\frac{1}{20^{2}}

$$

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Using Sigma Notation to Write a Sum In

Exercises $79-88$ , use sigma notation to write the sum.

$$

\frac{1}{1 \cdot 3}+\frac{1}{2 \cdot 4}+\frac{1}{3 \cdot 5}+\dots+\frac{1}{10 \cdot 12}

$$

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Using Sigma Notation to Write a Sum In

Exercises $79-88$ , use sigma notation to write the sum.

$$

\frac{1}{4}+\frac{3}{8}+\frac{7}{16}+\frac{15}{32}+\frac{31}{64}

$$

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Using Sigma Notation to Write a Sum In

Exercises $79-88$ , use sigma notation to write the sum.

$$

\frac{1}{2}+\frac{2}{4}+\frac{6}{8}+\frac{24}{16}+\frac{120}{32}+\frac{720}{64}

$$

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Finding a Partial Sum of a Series In Exercises

$89-92,$ find the indicated partial sum of the series.

$$

\sum_{i=1}^{\infty} 5\left(\frac{1}{2}\right)^{i}

$$ Fourth partial sum

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Finding a Partial Sum of a Series In Exercises

$89-92,$ find the indicated partial sum of the series.

$$\sum_{i=1}^{\infty} 2\left(\frac{1}{3}\right)^{i}$$ Fifth partial sum

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Finding a Partial Sum of a Series In Exercises

$89-92,$ find the indicated partial sum of the series.

$$\sum_{n=1}^{\infty} 4\left(-\frac{1}{2}\right)^{n}$$ Third partial sum

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Finding a Partial Sum of a Series In Exercises

$89-92,$ find the indicated partial sum of the series.

$$\sum_{n=1}^{\infty} 8\left(-\frac{1}{4}\right)^{n}$$ Fourth partial sum

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Finding the Sum of an Infinite Series

Exercises $93-96$ , find the sum of the infinite series.

$$

\sum_{i=1}^{\infty} \frac{6}{10^{i}}

$$

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Finding the Sum of an Infinite Series

Exercises $93-96$ , find the sum of the infinite series.

$$

\sum_{k=1}^{\infty}\left(\frac{1}{10}\right)^{k}

$$

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Finding the Sum of an Infinite Series

Exercises $93-96$ , find the sum of the infinite series.

$$

\sum_{k=1}^{\infty} 7\left(\frac{1}{10}\right)^{k}

$$

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Finding the Sum of an Infinite Series

Exercises $93-96$ , find the sum of the infinite series.

$$

\sum_{i=1}^{\infty} \frac{2}{10^{i}}

$$

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Compound Interest An investor deposits $\$ 10,000$ in

an account that earns 3.5$\%$ interest compounded quarterly.

The balance in the account after $n$ quarters is given by

$$

A_{n}=10,000\left(1+\frac{0.035}{4}\right)^{n}, \quad n=1,2,3, \ldots

$$

(a) Write the first eight terms of the sequence.

(b) Find the balance in the account after 10 years by

computing the 40 th term of the sequence.

(c) Is the balance after 20 years twice the balance after

10 years? Explain.

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The numbers $a_{n}$ (in thousands) of AIDS cases reported

from 2003 through 2010 can be approximated by $a_{n}=-0.0126 n^{3}+0.391 n^{2}-4.21 n+48.5$

$n=3,4, \ldots, 10$

where $n$ is the

year, with $n=3$

corresponding to

$2003 .$ (Source:

U.S. Centers for

Disease Control

and Prevention)

(a) Write the terms of this finite sequence. Use a

graphing utility to construct a bar graph that

represents the sequence.

(b) What does the graph in part (a) say about reported

cases of AIDS?

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True or False? In Exercises 99 and $100,$ determine

whether the statement is true or false. Justify your answer.

$$

\sum_{i=1}^{4}\left(i^{2}+2 i\right)=\sum_{i=1}^{4} i^{2}+2 \sum_{i=1}^{4} i

$$

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True or False? In Exercises 99 and $100,$ determine

whether the statement is true or false. Justify your answer.

True or False? In Exercises 99 and $100,$ determine

whether the statement is true or false. Justify your answer.

$$

\sum_{j=1}^{4} 2^{j}=\sum_{j=3}^{6} 2^{j-2}

$$

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Arithmetic Mean In Exercises $101-103,$ use the

following definition of the arithmetic mean $\overline{x}$ of a set of $n$

measurements $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$

$$

\overline{x}=\frac{1}{n} \sum_{i=1}^{n} x_{i}

$$

Find the arithmetic mean of the six checking account

balances $\$ 327.15, \quad \$ 785.69, \quad \$ 433.04, \quad \$ 265.38$

$\$ 604.12,$ and $\$ 590.30 .$ Use the statistical capabilities

of a graphing utility to verify your result.

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Arithmetic Mean In Exercises $101-103,$ use the

following definition of the arithmetic mean $\overline{x}$ of a set of $n$

measurements $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$

$$

\overline{x}=\frac{1}{n} \sum_{i=1}^{n} x_{i}

$$

Proof Prove that $$\sum_{i=1}^{n}\left(x_{i}-\overline{x}\right)=0$$

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Arithmetic Mean In Exercises $101-103,$ use the

following definition of the arithmetic mean $\overline{x}$ of a set of $n$

measurements $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$

$$

\overline{x}=\frac{1}{n} \sum_{i=1}^{n} x_{i}

$$

Proof Prove that $$\sum_{i=1}^{n}\left(x_{i}-\overline{x}\right)^{2}=\sum_{i=1}^{n} x_{i}^{2}-\frac{1}{n}\left(\sum_{i=1}^{n} x_{i}\right)^{2}$$

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HOW DO YOU SEE IT? The graph

represents the first 10 terms of a sequence.

Complete each expression for the apparent $n$ th

term $a_{n}$ of the sequence. Which expressions

are appropriate to represent the cost $a_{n}$ to buy

$n \mathrm{MP} 3$ songs at a cost of $\$ 1$ per song?

Explain.

$$

a_{n}=1

$$

$$

a_{n}=\frac{!}{(n-1) !}

$$

$$

a_{n}=\sum_{k=1}^{n}

$$

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Finding the Terms of a Sequence In Exercises 105

and $106,$ find the first five terms of the sequence.

$$

a_{n}=\frac{x^{n}}{n !}

$$

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Finding the Terms of a Sequence In Exercises 105

and $106,$ find the first five terms of the sequence.

$$

a_{n}=\frac{(-1)^{n} x^{2 n+1}}{2 n+1}

$$

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Cube $A 3 \times 3 \times 3$ cube is made up of 27 unit cubes

(a unit cube has a length, width, and height of 1 unit),

and only the faces of each cube that are visible are

painted blue, as shown in the figure.

(a) Complete the table to determine how many unit

cubes of the $3 \times 3 \times 3$ cube have 0 blue faces,

1 blue face, 2 blue faces, and 3 blue faces.

(b) Repeat part (a) for a $4 \times 4 \times 4$ cube, a $5 \times 5 \times 5$

cube, and $66 \times 6 \times 6$ cube.

(c) What type of pattern do you observe?

(d) Write formulas you could use to repeat part (a) for

$\quad$ an $n \times n \times n$ cube.

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