Chapter 9

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

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

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

For the following exercises, express each geometric sum using summation notation. \(-\frac{1}{6}+\frac{1}{12}-\frac{1}{24}+\ldots+\frac{1}{768}\)

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. \(a_{n}=\left\\{\begin{array}{ll}(2 n+1)^{2} & \text { if } n \text { is divisible by } 4 \\ \frac{2}{n} & \text { if } n \text { is not divisible by } 4\end{array}\right.\)

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

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

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

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, 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 \(a_{14}=28\).

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

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

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

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

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

For the following exercises, find the specified term for the geometric sequence, given the first four terms. \(a_{n}=\\{-1,2,-4,8, \ldots\\} .\) Find \(a_{12}\)

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 \(a_{15}=42\).

For the following exercises, write the first eight terms of the piecewise sequence. \(a_{n}=\left\\{\begin{array}{ll}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, four coins are tossed. Find the probability of tossing four heads or four tails.

For the following exercises, use the Binomial Theorem to expand each binomial. $$ \left(x^{-1}+2 y^{-1}\right)^{4} $$

For the following exercises, compute the value of the expression. $$ C(12,4) $$

For the following exercises, use the formula for the sum of the first \(n\) terms of each geometric sequence, and then state the indicated sum. \(\sum_{a=1}^{11} 64 \cdot 0.2^{a-1}\)

For the following exercises, find the specified term for the geometric sequence, given the first four terms. \(a_{n}=\left\\{-2, \frac{2}{3},-\frac{2}{9}, \frac{2}{27}, \ldots\right\\} .\) Find \(a_{7}\)

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 \(a_{8}=40\) and \(a_{23}=115\).

For the following exercises, write an explicit formula for each sequence. \(4,7,12,19,28, \ldots\)

For the following exercises, four coins are tossed. Find the probability of tossing all tails.

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

For the following exercises, compute the value of the expression. $$ C(26,3) $$

For the following exercises, determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason. \(12+18+24+30+\ldots\)

For the following exercises, write the first five terms of the geometric sequence. \(a_{1}=-486, \quad a_{n}=-\frac{1}{3} a_{n-1}\)

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 \(a_{9}=54\) and \(a_{17}=102\).

For the following exercises, write an explicit formula for each sequence. \(-4,2,-10,14,-34, \ldots\)

For the following exercises, four coins are tossed. Find the probability of tossing not all tails.

For the following exercises, use the Binomial Theorem to write the first three terms of each binomial. $$ (a+b)^{17} $$

For the following exercises, compute the value of the expression. $$ C(7,6) $$

For the following exercises, determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason. \(2+1.6+1.28+1.024+\ldots\)

For the following exercises, write the first five terms of the geometric sequence. \(a_{1}=7, \quad a_{n}=0.2 a_{n-1}\)

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 \(a_{11}=11\) and \(a_{21}=16\).

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

For the following exercises, use the Binomial Theorem to write the first three terms of each binomial. $$ (x-1)^{18} $$

For the following exercises, compute the value of the expression. $$ C(10,3) $$

For the following exercises, determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason. \(\sum_{m=1}^{\infty} 4^{m-1}\)

For the following exercises, write a recursive formula for each geometric sequence. \(a_{n}=\\{-1,5,-25,125, \ldots\\}\)

For the following exercises, find the specified term given two terms from an arithmetic sequence. \(a_{1}=33\) and \(a_{7}=-15\). Find \(a_{4}\).

For the following exercises, write an explicit formula for each sequence. \(0, \frac{1-e^{1}}{1+e^{2}}, \frac{1-e^{2}}{1+e^{3}}, \frac{1-e^{3}}{1+e^{4}}, \frac{1-e^{4}}{1+e^{5}}, \ldots\)

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

For the following exercises, use the Binomial Theorem to write the first three terms of each binomial. $$ (a-2 b)^{15} $$

For the following exercises, find the number of subsets in each given set. $$ \\{1,2,3,4,5,6,7,8,9,10\\} $$

For the following exercises, determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason. \(\sum_{\infty}^{k=1}-\left(-\frac{1}{2}\right)^{k-1}\)