Chapter 8

Counting Strings Count the number of five-letter strings that can be formed with the given letters, assuming a letter can be used more than once. \(\mathrm{W}, \mathrm{X}, \mathrm{Y}, \mathrm{Z}\)

For the given \(a_{n},\) calculate \(S_{5}.\) $$ a_{n}=4 n+1 $$

Calculate the number of distinguishable strings that can be formed with the given number of a's and b's. Four \(a^{\prime}\) 's, four \(b\) 's

Find the probability of each event. Rolling a 2 with a fair die

Use mathematical induction to prove the statement. Assume that \(n\) is a positive integer. $$ \frac{4}{5}+\frac{4}{5^{2}}+\frac{4}{5^{3}}+\dots+\frac{4}{5^{n}}=1-\frac{1}{5^{n}} $$

Find the first four terms of the sequence. \(a_{n}=2^{n}+n^{2}\)

Counting Strings Count the number of five-letter strings that can be formed with the given letters, assuming a letter can be used more than once. \(\mathbf{D}, \mathbf{E}, \mathbf{F}, \mathbf{G}, \mathbf{H}\)

For the given \(a_{n},\) calculate \(S_{5}.\) $$ a_{n}=n^{2}+1 $$

Calculate the number of distinguishable strings that can be formed with the given number of a's and b's. One \(a,\) five \(b\) 's

Find the probability of each event. Rolling a 5 or 6 with a fair die

Use mathematical induction to prove the statement. Assume that \(n\) is a positive integer. $$ \frac{1}{2}+\frac{1}{2^{2}}+\frac{1}{2^{3}}+\dots+\frac{1}{2^{n}}=1-\frac{1}{2^{n}} $$

Find the first four terms of the sequence. \(a_{n}=\frac{1}{n}+\frac{1}{3 n}\)

Calculate the number of distinguishable strings that can be formed with the given number of a's and b's. Five \(a^{\prime} 8,\) no \(b^{\prime}\) s

Find the probability of each event. Guessing the correct answer to a true-false question

Use mathematical induction to prove the statement. Assume that \(n\) is a positive integer. $$ \frac{1}{1 \cdot 4}+\frac{1}{4 \cdot 7} \cdot+\dots+\frac{1}{(3 n-2)(3 n+1)}=\frac{n}{3 n+1} $$

Counting Strings Count the number of strings that can be formed with the given letters, assuming each letter is used exactly once. \(A, B\)

For the given \(a_{n},\) calculate \(S_{5}.\) $$ a_{\mathrm{n}}=\frac{n}{n+1} $$

Calculate the number of distinguishable strings that can be formed with the given number of a's and b's. No \(a^{\prime \prime}\) s, three \(b^{\prime}\) s

Find the probability of each event. Guessing the correct answer to a multiple-choice question with five choices

Use mathematical induction to prove the statement. Assume that \(n\) is a positive integer. $$ x^{2 n}+x^{2 n-1} y+\cdots+x y^{2 n-1}+y^{2 n}=\frac{x^{2 n+1}-y^{2 n+1}}{x-y} $$

Counting Strings Count the number of strings that can be formed with the given letters, assuming each letter is used exactly once. \(A, B, C\)

For the given \(a_{n},\) calculate \(S_{5}.\) $$ a_{n}=\frac{1}{2 n} $$

Find the probability of each event. Randomly drawing a king from a standard deck of 52 cards

Find all positive integers \(n\) for which the given statement is not true. $$ 3^{n}>6 n $$

Complete the following for the recursively defined sequence. (a) Find the first four terms. (b) Graph these terms. \(a_{n}=2 a_{n-1} ; a_{1}=1\)

Counting Strings Count the number of strings that can be formed with the given letters, assuming each letter is used exactly once. \(\mathrm{W}, \mathrm{X}, \mathrm{Y}, \mathrm{Z}\)

Use a formula to find the sum of the arithmetic series. $$ 3+5+7+9+11+13+15+17 $$

Calculate the number of distinguishable strings that can be formed with the given number of a's and b's. Four \(a^{\prime}\) s, two b's

Find the probability of each event. Randomly drawing a club from a standard deck of 52 cards

Find all positive integers \(n\) for which the given statement is not true. $$ 3^{n}>2 n+1 $$

Complete the following for the recursively defined sequence. (a) Find the first four terms. (b) Graph these terms. \(a_{n}=a_{n-1}+5 ; a_{1}=-4\)

Counting Strings Count the number of strings that can be formed with the given letters, assuming each letter is used exactly once. \(\mathrm{V}, \mathrm{W}, \mathrm{X}, \mathrm{Y}, \mathrm{Z}\)

Use a formula to find the sum of the arithmetic series. $$ 7.5+6+4.5+3+1.5+0+(-1.5) $$

Use the binomial theorem to expand each expression. $$ (x+y)^{2} $$

Find all positive integers \(n\) for which the given statement is not true. $$ 2^{n}>n^{2} $$

Complete the following for the recursively defined sequence. (a) Find the first four terms. (b) Graph these terms. \(a_{n}=a_{n-1}+3 ; a_{1}=-3\)

Combination Lock A briefcase has two locks. The combination to each lock consists of a three-digit number, in which digits may be repeated. See the figure. How many combinations are possible? (Hint: The word combination is a misnomer. Lock combinations are permutations in which the arrangement of the numbers is important.)

Use a formula to find the sum of the arithmetic series. $$ 1+2+3+4+\dots+50 $$

Use the binomial theorem to expand each expression. $$ (x+y)^{4} $$

Complete the following for the recursively defined sequence. (a) Find the first four terms. (b) Graph these terms. \(a_{n}=2 a_{n-1}+1 ; a_{1}=1\)

\(\mathbf{A}\) typical combination for a padlock consists of three numbers from 0 to 39. Count the combinations possible with this type of lock if a number may be repeated.

Use a formula to find the sum of the arithmetic series. $$ 1+3+5+7+\dots+97 $$

Use the binomial theorem to expand each expression. $$ (m+2)^{3} $$

Prove the statement by mathematical induction. \(\left(a^{m}\right)^{n}=a^{m n}(\text { Assume } a \text { and } m\) are constants.)

Complete the following for the recursively defined sequence. (a) Find the first four terms. (b) Graph these terms. \(a_{n}=3 a_{n-1}-1 ; a_{1}=2\)

Garage Door Openers The code for some garage door openers consists of 12 electrical switches that can be set to either 0 or 1 by the owner. With this type of opener, how many codes are possible? (Source: Promax.)

Use a formula to find the sum of the arithmetic series. $$ -7+(-4)+(-1)+2+5+\dots+98+101 $$

Use the binomial theorem to expand each expression. $$ (m+2 n)^{5} $$

Complete the following for the recursively defined sequence. (a) Find the first four terms. (b) Graph these terms. \(a_{n}=\frac{1}{2} a_{n-1} ; a_{1}=16\)

To win the jackpot in a lottery game, a person must pick three numbers from 0 to 9 in the correct order. If a number can be repeated, how many ways are there to play the game?