Chapter 9

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

Use the formula for the sum of the fi st \(n\) terms of each geometric sequence, and then state the indicated sum. $$ \sum_{a=1}^{11} 64 \cdot 0.2^{a-1} $$

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

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

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

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

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 \(a_{8}=40\) 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, find the specified term for the geometric sequence, given the first four terms. $$a_{n}=\left\\{-2, \frac{2}{3},-\frac{2}{9}, \frac{2}{27}, \ldots .\right\\} \text { Find } a_{7}$$

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 \(a_{23}=115 .\)

Use the formula for the sum of the first n terms of each geometric sequence, and then state the indicated sum. $$\sum_{a=1}^{11} 64 \cdot 0.2^{a-1}$$

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

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

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

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

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

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 \(a_{17}=102 .\)

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

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

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

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

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

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 $$ a_{11}=11 \text { and } a_{21}=16 $$

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

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

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. $$ a_{1}=33 \text { and } a_{7}=-15 . \text { Find } a_{4} $$

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

Determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason. $$\sum_{m=1}^{\infty} 4^{m-1}$$

Write an explicit formula for each sequence. $$0, \frac{1-e^{1}}{1+e^{2}}, \frac{1-e^{2}}{1+e^{3}}, \frac{1-e^{3}}{1+e^{4}}, \frac{1-e^{4}}{1+e^{5}}, \ldots$$

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

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

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

For the following exercises, write a recursive formula for each geometric sequence. $$ a_{n}=\\{-32,-16,-8,-4, \ldots\\} $$

For the following exercises, find the specified term given two terms from an arithmetic sequence. $$ a_{3}=-17.1 \text { and } a_{10}=-15.7 . \text { Find } a_{21} $$.

For the following exercises, write an explicit formula for each sequence. $$ 1,-\frac{1}{2}, \frac{1}{4},-\frac{1}{8}, \frac{1}{16}, \ldots $$

Write an explicit formula for each sequence. $$1,-\frac{1}{2}, \frac{1}{4},-\frac{1}{8}, \frac{1}{16}, \dots$$

For the following exercises, one card is drawn from a standard deck of 52 cards. Find the probability of drawing the following: A club

Use the following scenario. Javier makes monthly deposits into a savings account. He opened the account with an initial deposit of \(\$ 50 .\) Each month thereafter he increased the previous deposit amount by \(\$ 20\). Graph the arithmetic sequence showing one year of Javier's deposits.

For the following exercises, find the number of subsets in each given set. $$ \\{a, b, c, \ldots, z\\} $$

For the following exercises, use the Binomial Theorem to write the first three terms of each binomial. $$ (x-2 y)^{8} $$

For the following exercises, write a recursive formula for each geometric sequence. $$ a_{n}=\\{14,56,224,896, \ldots\\} $$

For the following exercises, use the recursive formula to write the first five terms of the arithmetic sequence. $$ a_{1}=39 ; a_{n}=a_{n-1}-3 $$

For the following exercises, write the first five terms of the sequence. $$ a_{1}=9, a_{n}=a_{n-1}+n $$

For the following exercises, use the following scenario. Javier makes monthly deposits into a savings account. He opened the account with an initial deposit of \(\$ 50 .\) Each month thereafter he increased the previous deposit amount by \(\mathrm{s} 20 .\) Graph the arithmetic sequence showing one year of Javier’s deposits.

Write the first five terms of the sequence. $$a_{1}=9, a_{n}=a_{n-1}+n$$