Chapter 11

write the first four terms of each sequence whose general term is given. $$ a_{n}=(-1)^{n+1}(n+4) $$

A statement \(S_{n}\) about the positive integers is given. Write statements \(S_{k}\) and \(S_{k+1},\) simplifying statement \(S_{k+1}\) completely. \(S_{n}: 2+7+12+\cdots+(5 n-3)=\frac{n(5 n-1)}{2}\)

Write the first six terms of each arithmetic sequence. $$ a_{1}=\frac{3}{4}, d=-\frac{1}{4} $$

Shown again is the table indicating the marital status of the U.S. population in 2010. Numbers in the table are expressed in millions. Use the data in the table to solve Exercises \(1-10 .\) Express probabilities as simplified fractions and as decimals rounded to the nearest hundredth $$ \begin{array}{llllll} {} & {} & {\text { Never}} \\ {} & {\text {Married}} & {\text { Married }} & {\text { Divorced }} & {\text { Widowed }} & {\text { Total }} \\ \hline \text { Male } & {65} & {40} & {10} & {3} & {118} \\ \hline \text { Female } & {65} & {34} & {14} & {11} & {124} \\ \hline \text { Total } & {130} & {74} & {24} & {14} & {242} \end{array} $$ If one person is randomly selected from the population described in the table, find the probability, expressed as a simplified fraction and as a decimal to the nearest hundredth, that the person $$\text{Among men, find the probability of selecting a married person.}$$

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, \(r .\) Find \(a_{8}\) when \(a_{1}=6, r=2\)

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (x+2)^{3} $$

Use the formula for \(_{n} C,\) to evaluate each expression. \(_{9} C_{5}\)

write the first four terms of each sequence whose general term is given. $$ a_{n}=\frac{2 n}{n+4} $$

Write the first six terms of each arithmetic sequence. $$ a_{n}=a_{n-1}+6, a_{1}=-9 $$

Shown again is the table indicating the marital status of the U.S. population in 2010. Numbers in the table are expressed in millions. Use the data in the table to solve Exercises \(1-10 .\) Express probabilities as simplified fractions and as decimals rounded to the nearest hundredth $$ \begin{array}{llllll} {} & {} & {\text { Never}} \\ {} & {\text {Married}} & {\text { Married }} & {\text { Divorced }} & {\text { Widowed }} & {\text { Total }} \\ \hline \text { Male } & {65} & {40} & {10} & {3} & {118} \\ \hline \text { Female } & {65} & {34} & {14} & {11} & {124} \\ \hline \text { Total } & {130} & {74} & {24} & {14} & {242} \end{array} $$ If one person is randomly selected from the population described in the table, find the probability, expressed as a simplified fraction and as a decimal to the nearest hundredth, that the person $$\text{Among women, find the probability of selecting a married person.}$$

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, \(r .\) Find \(a_{8}\) when \(a_{1}=5, r=3\)

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

Use the formula for \(_{n} C,\) to evaluate each expression. \(_{10} C_{6}\)

write the first four terms of each sequence whose general term is given. $$ a_{n}=\frac{3 n}{n+5} $$

A statement \(S_{n}\) about the positive integers is given. Write statements \(S_{k}\) and \(S_{k+1},\) simplifying statement \(S_{k+1}\) completely. \(S_{n}: 2\) is a factor of \(n^{2}-n\)

Write the first six terms of each arithmetic sequence. $$ a_{n}=a_{n-1}+4, a_{1}=-7 $$

A die is rolled. Find the probability of getting $$a 4$$

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, \(r .\) Find \(a_{12}\) when $a_{1}=5, r=-2

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

write the first four terms of each sequence whose general term is given. $$ a_{n}=\frac{(-1)^{n+1}}{2^{n}-1} $$

Use mathematical induction to prove that each statement is true for every positive integer n. \(4+8+12+\dots+4 n=2 n(n+1)\)

Write the first six terms of each arithmetic sequence. $$ a_{n}=a_{n-1}-10, a_{1}=30 $$

A die is rolled. Find the probability of getting $$ a 5 $$

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, \(r .\) Find \(a_{12}\) when \(a_{1}=4, r=-2\)

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

Use the formula for \(_{n} C,\) to evaluate each expression. \(_{12} C_{5}\)

write the first four terms of each sequence whose general term is given. $$ a_{n}=\frac{(-1)^{n+1}}{2^{n}+1} $$

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

Write the first six terms of each arithmetic sequence. $$ a_{n}=a_{n-1}-20, a_{1}=50 $$

A die is rolled. Find the probability of getting $$\text {an odd number.}$$

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, \(r .\) Find \(a_{40}\) when \(a_{1}=1000, r=-\frac{1}{2}\)

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (5 x-1)^{3} $$

Use the formula for \(_{n} C,\) to evaluate each expression. \(_{7} C_{7}\)

are defined using recursion formulas. Write the first four terms of each sequence. $$ a_{1}=7 \text { and } a_{n}=a_{n-1}+5 \text { for } n \geq 2 $$

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

Write the first six terms of each arithmetic sequence. $$ a_{n}=a_{n-1}-0.4, a_{1}=1.6 $$

A die is rolled. Find the probability of getting $$\text {a number greater than 3.}$$

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, \(r .\) Find \(a_{30}\) when \(a_{1}=8000, r=-\frac{1}{2}\)

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

Use the formula for \(_{n} C,\) to evaluate each expression. \(_{4} \mathrm{C}_{4}\)

are defined using recursion formulas. Write the first four terms of each sequence. $$ a_{1}=12 \text { and } a_{n}=a_{n-1}+4 \text { for } n \geq 2 $$

Use mathematical induction to prove that each statement is true for every positive integer n. \(3+6+9+\cdots+3 n=\frac{3 n(n+1)}{2}\)

Write the first six terms of each arithmetic sequence. $$ a_{n}=a_{n-1}-0.3, a_{1}=-1.7 $$

A die is rolled. Find the probability of getting $$\text{a number greater than 4.}$$

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, \(r .\) Find \(a_{8}\) when \(a_{1}=1,000,000, r=0.1\)

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

Use the formula for \(_{n} C,\) to evaluate each expression. \(_{5} C_{0}\)

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

Use mathematical induction to prove that each statement is true for every positive integer n. \(3+7+11+\cdots+(4 n-1)=n(2 n+1)\)

Find the indicated term of the arithmetic sequence with first term, \(a_{1},\) and common difference, \(d .\) Find \(a_{6}\) when \(a_{1}=13, d=4\)