Chapter 11

A die is rolled. Find the probability of getting $$\text{a number greater than 7.}$$

In Exercises \(9-16,\) use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of each sequence with the given first term, \(a_{1},\) and common ratio, \(r .\) $$ \text { Find } a_{8} \text { when } a_{1}=40,000, r=0.1 $$

$$ (3 x+1)^{4} $$

Use the formula for \(_{n} C,\) to evaluate each expression. \(_{6} C_{0}\)

are defined using recursion formulas. Write the first four terms of each sequence. $$ a_{1}=2 \text { and } a_{n}=5 a_{n-1} \text { for } n \geq 2 $$

Use mathematical induction to prove that each statement is true for every positive integer n. \(2+7+12+\cdots+(5 n-3)=\frac{n(5 n-1)}{2}\)

Find the indicated term of the arithmetic sequence with first term, \(a_{1},\) and common difference, \(d .\) Find \(a_{16}\) when \(a_{1}=9, d=2\)

You are dealt one card from a standard 52-card deck. Find the probability of being dealt $$\text{a queen.}$$

In Exercises \(17-24,\) write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7},\) the seventh term of the sequence. $$ 3,12,48,192, \dots $$

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ \left(x^{2}+2 y\right)^{4} $$

A medical researcher needs 6 people to test the effectiveness of an experimental drug. If 13 people have volunteered for the test, in how many ways can 6 people be selected?

are defined using recursion formulas. Write the first four terms of each sequence. $$ a_{1}=4 \text { and } a_{n}=2 a_{n-1}+3 \text { for } n \geq 2 $$

Use mathematical induction to prove that each statement is true for every positive integer n. \(1+2+2^{2}+\cdots+2^{n-1}=2^{n}-1\)

Find the indicated term of the arithmetic sequence with first term, \(a_{1},\) and common difference, \(d .\) Find \(a_{s 0}\) when \(a_{1}=7, d=5\)

You are dealt one card from a standard 52-card deck. Find the probability of being dealt $$\text{a diamond.}$$

In Exercises \(17-24,\) write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7},\) the seventh term of the sequence. $$ 3,15,75,375, \dots $$

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ \left(x^{2}+y\right)^{4} $$

Fifty people purchase raffle tickets. Three winning tickets are selected at random. If first prize is \(\$ 1000,\) second prize is \(\$ 500,\) and third prize is \(\$ 100,\) in how many different ways can the prizes be awarded?

are defined using recursion formulas. Write the first four terms of each sequence. $$ a_{1}=5 \text { and } a_{n}=3 a_{n-1}-1 \text { for } n \geq 2 $$

Use mathematical induction to prove that each statement is true for every positive integer n. \(1+3+3^{2}+\cdots+3^{n-1}=\frac{3^{n}-1}{2}\)

Find the indicated term of the arithmetic sequence with first term, \(a_{1},\) and common difference, \(d .\) Find \(a_{60}\) when \(a_{1}=8, d=6\)

You are dealt one card from a standard 52-card deck. Find the probability of being dealt $$\text{a picture card.}$$

In Exercises \(17-24,\) write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7},\) the seventh term of the sequence. $$ 18,6,2, \frac{2}{3}, \dots $$

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (y-3)^{4} $$

How many different four-letter passwords can be formed from the letters \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}, \mathrm{F},\) and \(\mathrm{G}\) if no repetition of letters is allowed?

the general term of a sequence is given and involves a factorial. Write the first four terms of each sequence. $$ a_{n}=\frac{n^{2}}{n !} $$

Use mathematical induction to prove that each statement is true for every positive integer n. \(2+4+8+\cdots+2^{n}=2^{n+1}-2\)

Find the indicated term of the arithmetic sequence with first term, \(a_{1},\) and common difference, \(d .\) Find \(a_{200}\) when \(a_{1}=-40, d=5\)

You are dealt one card from a standard 52-card deck. Find the probability of being dealt $$\text{a card greater than 3 and less than 7.}$$

In Exercises \(17-24,\) write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7},\) the seventh term of the sequence. $$ 12,6,3, \frac{3}{2}, \dots $$

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (y-4)^{4} $$

Fifty people purchase raffle tickets. Three winning tickets are selected at random. If each prize is \(\$ 500,\) in how many different ways can the prizes be awarded?

the general term of a sequence is given and involves a factorial. Write the first four terms of each sequence. $$ a_{n}=\frac{(n+1) !}{n^{2}} $$

Use mathematical induction to prove that each statement is true for every positive integer n. \(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots+\frac{1}{2^{n}}=1-\frac{1}{2^{n}}\)

Find the indicated term of the arithmetic sequence with first term, \(a_{1},\) and common difference, \(d .\) Find \(a_{150}\) when \(a_{1}=-60, d=5\)

A fair coin is tossed two times in succession. The sample space of equally likely outcomes is \(\\{H H, H T, T H, T T\\} .\) Find the probability of getting $$\text{two heads.}$$

In Exercises \(17-24,\) write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7},\) the seventh term of the sequence. $$ 1.5,-3,6,-12, \dots $$

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ \left(2 x^{3}-1\right)^{4} $$

Evaluate each expression. \(\frac{7 P_{3}}{3 !}-_{7} C_{3}\)

the general term of a sequence is given and involves a factorial. Write the first four terms of each sequence. $$ a_{n}=2(n+1) ! $$

Use mathematical induction to prove that each statement is true for every positive integer n. \(1 \cdot 2+2 \cdot 3+3 \cdot 4+\cdots+n(n+1)=\frac{n(n+1)(n+2)}{3}\)

Find the indicated term of the arithmetic sequence with first term, \(a_{1},\) and common difference, \(d .\) Find \(a_{60}\) when \(a_{1}=35, d=-3\)

A fair coin is tossed two times in succession. The sample space of equally likely outcomes is \(\\{H H, H T, T H, T T\\} .\) Find the probability of getting $$\text{the same outcome on each toss.}$$

In Exercises \(17-24,\) write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7},\) the seventh term of the sequence. $$ 5,-1, \frac{1}{5},-\frac{1}{25}, \dots $$

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ \left(2 x^{5}-1\right)^{4} $$

the general term of a sequence is given and involves a factorial. Write the first four terms of each sequence. $$ a_{n}=-2(n-1) ! $$

Use mathematical induction to prove that each statement is true for every positive integer n. \(1 \cdot 3+2 \cdot 4+3 \cdot 5+\cdots+n(n+2)=\frac{n(n+1)(2 n+7)}{6}\)

Find the indicated term of the arithmetic sequence with first term, \(a_{1},\) and common difference, \(d .\) Find \(a_{70}\) when \(a_{1}=-32, d=4\)

You select a family with three children. If \(M\) represents a male child and \(F\) a female child, the sample space of equally likely outcomes is \(\\{M M M, M M F, M F M, M F F, F M M FMF, FFM, FFF\)} - Find the probability of selecting a family with $$\text{at least one male child.}$$

In Exercises \(17-24,\) write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7},\) the seventh term of the sequence. $$ 0.0004,-0.004,0.04,-0.4, \ldots $$