Chapter 9

For the following exercises, write the first four terms of the sequence. $$ a_{n}=(-10)^{n}+1 $$

Write the first four terms of the sequence. $$a_{n}=(-10)^{n}+1$$

For the following exercises, two coins are tossed. Find the probability of tossing two heads.

For the following exercises, compute the value of the expression. $$ P(5,2) $$

For the following exercises, use the Binomial Theorem to expand each binomial. $$ (3 a+2 b)^{3} $$

For the following exercises, write the first five terms of the geometric sequence, given the first term and common ratio. $$ a_{1}=5, r=\frac{1}{5} $$

For the following exercises, find the specified term for the arithmetic sequence given the first term and common difference. First term is \(4,\) common difference is \(5,\) fi \(\mathrm{d}\) the \(4^{\text {th }}\) term.

For the following exercises, write the first four terms of the sequence. $$ a_{n}=-\left(\frac{4 \cdot(-5)^{n-1}}{5}\right) $$

Write the first four terms of the sequence. $$a_{n}=-\left(\frac{4 \cdot(-5)^{n-1}}{5}\right)$$

Express each geometric sum using summation notation. $$ 1+3+9+27+81+243+729+2187 $$

For the following exercises, two coins are tossed. Find the probability of tossing exactly one tail.

For the following exercises, compute the value of the expression. $$ P(8,4) $$

For the following exercises, use the Binomial Theorem to expand each binomial. $$ (2 x+3 y)^{4} $$

For the following exercises, write the first five terms of the geometric sequence, given any two terms. $$ a_{7}=64, a_{10}=512 $$

For the following exercises, find the specified term for the arithmetic sequence given the first term and common difference. First term is \(5,\) common difference is \(6,\) fi \(d\) the \(8^{\text {th }}\) term.

For the following exercises, write the first eight terms of the piecewise sequence. $$ a_{n}=\left\\{\begin{array}{l} (-2)^{n}-2 \text { if } n \text { is even } \\ (3)^{n-1} \text { if } n \text { is odd } \end{array}\right. $$

Write the first eight terms of the piecewise sequence. $$a_{n}=\left\\{\begin{array}{l}{(-2)^{n}-2 \text { if } n \text { is even }} \\\ {(3)^{n-1} \text { if } n \text { is odd }}\end{array}\right.$$

Express each geometric sum using summation notation. $$ 8+4+2+\ldots+0.125 $$

For the following exercises, two coins are tossed. Find the probability of tossing at least one tail.

For the following exercises, compute the value of the expression. $$ P(3,3) $$

For the following exercises, use the Binomial Theorem to expand each binomial. $$ (4 x+2 y)^{5} $$

For the following exercises, write the first five terms of the geometric sequence, given any two terms. $$ a_{6}=25, a_{8}=6.25 $$

For the following exercises, find the specified term for the arithmetic sequence given the first term and common difference. First term is 6 , common difference is 7 , fi d the \(6^{\text {th }}\) term.

For the following exercises, write the first eight terms of the piecewise sequence. $$ a_{n}=\left\\{\begin{array}{l} \frac{n^{2}}{2 n+1} \text { if } n \leq 5 \\ n^{2}-5 \text { if } n>5 \end{array}\right. $$

Write the first eight terms of the piecewise sequence. $$a_{n}=\left\\{\begin{array}{l}{\frac{n^{2}}{2 n+1} \text { if } n \leq 5} \\\ {n^{2}-5 \text { if } n>5}\end{array}\right.$$

For the following exercises, four coins are tossed. What is the sample space?

For the following exercises, compute the value of the expression. $$ P(9,6) $$

For the following exercises, use the Binomial Theorem to expand each binomial. $$ (3 x-2 y)^{4} $$

For the following exercises, find the specified term for the geometric sequence, given the first term and common ratio. The first term is 2 , and the common ratio is 3 . Find the \(5^{\text {th }}\) term.

For the following exercises, write the first eight terms of the piecewise sequence. $$ a_{n}=\left\\{\begin{array}{l} (2 n+1)^{2} \text { if } n \text { is divisible by } 4 \\ \frac{2}{n} \text { if } n \text { is not divisible by } 4 \end{array}\right. $$

Express each geometric sum using summation notation. \(-\frac{1}{6}+\frac{1}{12}-\frac{1}{24}+\ldots+\frac{1}{768}\)

Write the first eight terms of the piecewise sequence. $$a_{n}=\left\\{\begin{array}{l}{(2 n+1)^{2} \text { if } n \text { is divisible by } 4} \\ {\frac{2}{n} \text { if } n \text { is not divisible by } 4}\end{array}\right.$$

Use the formula for the sum of the fi st \(n\) terms of each geometric sequence, and then state the indicated sum. $$ 9+3+1+\frac{1}{3}+\frac{1}{9} $$

For the following exercises, four coins are tossed. Find the probability of tossing exactly two heads.

For the following exercises, compute the value of the expression. $$ P(11,5) $$

For the following exercises, use the Binomial Theorem to expand each binomial. $$ (4 x-3 y)^{5} $$

For the following exercises, find the specified term for the geometric sequence, given the first term and common ratio. The first term is 16 and the common ratio is \(-\frac{1}{3}\). Find the \(4^{\text {th }}\) term.

For the following exercises, fi d the fi st term given two terms from an arithmetic sequence. Find the fi st term or \(a_{1}\) of an arithmetic sequence if \(a_{6}=12\) and \(a_{14}=28\).

For the following exercises, write the first eight terms of the piecewise sequence. $$ a_{n}=\left\\{\begin{array}{l} -0.6 \cdot 5^{n-1} \text { if } n \text { is prime or } 1 \\ 2.5 \cdot(-2)^{n-1} \text { if } n \text { is composite } \end{array}\right. $$

For the following exercises, find the first term given two terms from an arithmetic sequence. Find the first term or \(a_{1}\) of an arithmetic sequence if \(a_{6}=12\) and \(a_{14}=28\)

Use the formula for the sum of the first n terms of each geometric sequence, and then state the indicated sum. \(9+3+1+\frac{1}{3}+\frac{1}{9}\)

Write the first eight terms of the piecewise sequence. $$a_{n}=\left\\{\begin{array}{l}{-0.6 \cdot 5^{n-1} \text { if } n \text { is prime or } 1} \\ {2.5 \cdot(-2)^{n-1} \text { if } n \text { is composite }}\end{array}\right.$$

Use the formula for the sum of the fi st \(n\) terms of each geometric sequence, and then state the indicated sum. $$ \sum_{n=1}^{9} 5 \cdot 2^{n-1} $$

For the following exercises, four coins are tossed. Find the probability of tossing exactly three heads.

For the following exercises, compute the value of the expression. $$ C(8,5) $$

For the following exercises, use the Binomial Theorem to expand each binomial. $$ \left(\frac{1}{x}+3 y\right)^{5} $$

For the following exercises, fi d the fi st term given two terms from an arithmetic sequence. Find the fi st term or \(a_{1}\) of an arithmetic sequence if \(a_{7}=21\) and \(a_{15}=42\).

For the following exercises, write the first eight terms of the piecewise sequence. $$ a_{n}=\left\\{\begin{array}{l} 4\left(n^{2}-2\right) \text { if } n \leq 3 \text { or } n>6 \\ \frac{n^{2}-2}{4} \text { if } 3

For the following exercises, find the first term given two terms from an arithmetic sequence. Find the first term or \(a_{1}\) of an arithmetic sequence if \(a_{7}=21\) and \(a_{15}=42 .\)

Use the formula for the sum of the first n terms of each geometric sequence, and then state the indicated sum. \(\sum_{n=1}^{9} 5 \cdot 2^{n-1}\)