Chapter 9

Consider the expansion of \((x+b)^{40} .\) What is the exponent of \(b\) in the \(k\) th term?

For the following exercises, write an explicit formula for each arithmetic sequence. $$ a=\\{1.8,3.6,5.4, \ldots\\} $$

For the following exercises, graph the first five terms of the indicated sequence $$ a_{n}=\frac{(-1)^{n}}{n}+n $$

A motorcycle shop has 10 choppers, 6 bobbers, and 5 café racers - different types of vintage motorcycles. How many ways can the shop choose 3 choppers, 5 bobbers, and 2 café racers for a weekend showcase?

Graph the first five terms of the indicated sequence. $$a_{n}=\frac{(-1)^{n}}{n}+n$$

For the following exercises, use this scenario: a bag of M\&Ms contains 12 blue, 6 brown, 10 orange, 8 yellow, 8 red, and 4 green M\&Ms. Reaching into the bag, a person grabs 5 M\&Ms. What is the probability of getting 4 blue \(M \& M s ?\)

Determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate. Deposit amount: \(\$ 450 ;\) total deposits: \(60 ;\) interest rate: \(4.5 \%\), compounded quarterly

A conductor needs 5 cellists and 5 violinists to play at a diplomatic event. To do this, he ranks the orchestra's 10 cellists and 16 violinists in order of musical proficie cy. What is the ratio of the total cellist rankings possible to the total violinist rankings possible?

For the following exercises, use the information provided to graph the first five terms of the geometric sequence. $$ a_{1}=1, r=\frac{1}{2} $$

For the following exercises, write an explicit formula for each arithmetic sequence. $$ a=\\{-18.1,-16.2,-14.3, \ldots\\} $$

For the following exercises, graph the first five terms of the indicated sequence $$ a_{n}=\left\\{\begin{array}{l} \frac{4+n}{2 n} \text { if } n \text { in even } \\ 3+n \text { if } n \text { is odd } \end{array}\right. $$

Find \(\left(\begin{array}{c}{n} \\\ {k-1}\end{array}\right)+\left(\begin{array}{c}{n} \\ {k}\end{array}\right)\) and write the answer as a binomial coefficient in the form \(\left(\begin{array}{l}{n} \\ {k}\end{array}\right)\) Prove it. Hint Use the fact that, for any integer \(p,\) such that \(p \geq 1\) \(p !=p(p-1) !\).

Graph the first five terms of the indicated sequence. $$a_{n}=\left\\{\begin{array}{l}{\frac{4+n}{2 n} \text { if } n \text { in even }} \\ {3+n \text { if } n \text { is odd }}\end{array}\right.$$

For the following exercises, use this scenario: a bag of M\&Ms contains 12 blue, 6 brown, 10 orange, 8 yellow, 8 red, and 4 green M\&Ms. Reaching into the bag, a person grabs 5 M\&Ms. What is the probability of getting 3 blue M\&Ms?

A motorcycle shop has 10 choppers, 6 bobbers, and 5 café racers-different types of vintage motorcycles. How many ways can the shop choose 3 choppers, 5 bobbers, and 2 café racers for a weekend showcase?

For the following exercises, use the information provided to graph the first five terms of the geometric sequence. $$ a_{1}=3, a_{n}=2 a_{n-1} $$

For the following exercises, write an explicit formula for each arithmetic sequence. $$ a=\\{15.8,18.5,21.2, \ldots\\} $$

For the following exercises, graph the first five terms of the indicated sequence $$ a_{1}=2, a_{n}=\left(-a_{n-1}+1\right)^{2} $$

A skateboard shop stocks 10 types of board decks, 3 types of trucks, and 4 types of wheels. How many different skateboards can be constructed?

Which expression cannot be expanded using the Binomial Theorem? Explain. a. \(\left(x^{2}-2 x+1\right)\) b. \((\sqrt{a}+4 \sqrt{a}-5)^{8}\) c. \(\left(x^{3}+2 y^{2}-z\right)^{5}\) d. \(\left(3 x^{2}-\sqrt{2 y^{3}}\right)^{12}\)

Determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate. Deposit amount: \(\$ 100 ;\) total deposits: \(120 ;\) interest rate: 10\(\%\) , compounded semi-annually

Graph the first five terms of the indicated sequence. $$a_{1}=2, a_{n}=\left(-a_{n-1}+1\right)^{2}$$

For the following exercises, use this scenario: a bag of M\&Ms contains 12 blue, 6 brown, 10 orange, 8 yellow, 8 red, and 4 green M\&Ms. Reaching into the bag, a person grabs 5 M\&Ms. What is the probability of getting no brown M\&Ms?

The sum of terms \(50-k^{2}\) from \(k=x\) through 7 is 115. What is \(x ?\)

A skateboard shop stocks 10 types of board decks, 3 types of trucks, and 4 types of wheels. How many different skateboards can be constructed?

For the following exercises, use the information provided to graph the first five terms of the geometric sequence. $$ a_{n}=27 \cdot 0.3^{n-1} $$

For the following exercises, write an explicit formula for each arithmetic sequence. $$ a=\left\\{\frac{1}{3},-\frac{4}{3},-3, \ldots\right\\} $$

Just-For-Kicks Sneaker Company offers an online customizing service. How many ways are there to design a custom pair of Just-For-Kicks sneakers if a customer can choose from a basic shoe up to 11 customizable options?

Use the following scenario for the exercises that follow: In the game of Keno, a player starts by selecting 20 numbers from the numbers 1 to 80 . After the player makes his selections, 20 winning numbers are randomly selected from numbers 1 to 80 . A win occurs if the player has correctly selected 3,4 , or 5 of the 20 winning numbers. (Round all answers to the nearest hundredth of a percent.) What is the percent chance that a player selects exactly 3 winning numbers?

Use recursive formulas to give two examples of geometric sequences whose \(3^{\text {rd }}\) terms are 200 .

For the following exercises, write an explicit formula for each arithmetic sequence. $$ a=\left\\{0, \frac{1}{3}, \frac{2}{3}, \ldots\right\\} $$

For the following exercises, graph the first five terms of the indicated sequence $$ a_{n}=\frac{(n+1) !}{(n-1) !} $$

A car wash offers the following optional services to the basic wash: clear coat wax, triple foam polish, undercarriage wash, rust inhibitor, wheel brightener, air freshener, and interior shampoo. How many washes are possible if any number of options can be added to the basic wash?

Graph the first five terms of the indicated sequence. $$a_{n}=\frac{(n+1) !}{(n-1) !}$$

Use the following scenario for the exercises that follow: In the game of Keno, a player starts by selecting 20 numbers from the numbers 1 to 80 . After the player makes his selections, 20 winning numbers are randomly selected from numbers 1 to 80 . A win occurs if the player has correctly selected 3,4 , or 5 of the 20 winning numbers. (Round all answers to the nearest hundredth of a percent.) What is the percent chance that a player selects exactly 4 winning numbers?

Find the smallest value of \(n\) such that \(\sum_{k=1}^{n}(3 k-5)>100\).

A car wash offers the following optional services to the basic wash: clear coat wax, triple foam polish, undercarriage wash, rust inhibitor, wheel brightener, air freshener, and interior shampoo. How many washes are possible if any number of options can be added to the basic wash?

For the following exercises, write an explicit formula for each arithmetic sequence. $$ a=\left\\{-5,-\frac{10}{3},-\frac{5}{3}, \ldots\right\\} $$

Use explicit formulas to give two examples of geometric sequences whose \(7^{\text { th }}\) terms are 1024 .

Use the following scenario for the exercises that follow: In the game of Keno, a player starts by selecting 20 numbers from the numbers 1 to 80 . After the player makes his selections, 20 winning numbers are randomly selected from numbers 1 to 80 . A win occurs if the player has correctly selected 3,4 , or 5 of the 20 winning numbers. (Round all answers to the nearest hundredth of a percent.) What is the percent chance that a player selects all 5 winning numbers?

How many terms must be added before the series \(-1-3-5-7 \ldots .\) has a sum less than \(-75 ?\)

Susan bought 20 plants to arrange along the border of her garden. How many distinct arrangements can she make if the plants are comprised of 6 tulips, 6 roses, and 8 daisies?

Find the \(5^{\text {th }}\) term of the geometric sequence \(\\{b, 4 b, 16 b, \ldots\\}\)

For the following exercises, find the number of terms in the given finite arithmetic sequence. $$ a=\\{3,-4,-11, \ldots,-60\\} $$

Write \(0 . \overline{65}\) as an infin te geometric series using summation notation. Then use the formula for fi ding the sum of an infin te geometric series to convert 0.65 to a fraction.

How many unique ways can a string of Christmas lights be arranged from 9 red, 10 green, 6 white, and 12 gold color bulbs?

For the following exercises, find the number of terms in the given finite arithmetic sequence. $$ a=\\{1.2,1.4,1.6, \ldots, 3.8\\} $$

Find the \(7^{\text { th }}\) term of the geometric sequence \(\\{64 a(-b), 32 a(-3 b), 16 a(-9 b), \ldots\\}\)

Write \(0 . \overline{65}\) as an infinite geometric series using summation notation. Then use the formula for finding the sum of an infinite geometric series to convert 0.65 to a fraction.

Use the following scenario for the exercises that follow: In the game of Keno, a player starts by selecting 20 numbers from the numbers 1 to 80 . After the player makes his selections, 20 winning numbers are randomly selected from numbers 1 to 80 . A win occurs if the player has correctly selected 3,4 , or 5 of the 20 winning numbers. (Round all answers to the nearest hundredth of a percent.) How much less is a player's chance of selecting 3 winning numbers than the chance of selecting either 4 or 5 winning numbers?