Chapter 8

Use a formula to find the sum of the arithmetic series. $$ 89+84+79+74+\dots+9+4 $$

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

Find the probability of the compound event. Tossing a coin twice with the outcomes of two tails

Prove the statement by mathematical induction. $$ 2^{n}>2 n \text { if } n \geq 3 $$

Complete the following for the recursively defined sequence. (a) Find the first four terms. (b) Graph these terms. \(a_{n}=a_{n-1}-a_{n-2} ; a_{1}=2, a_{2}=5\)

Call letters for a radio station usually begin with either a \(\mathrm{K}\) or a \(\mathrm{W}\), followed by three letters. In \(2005,\) there were \(13,517\) radio stations on the air. Is there any shortage of call letters for new radio stations?

Use a formula to find the sum of the arithmetic series. The first 40 terms of the series defined by \(a_{n}=5 n\)

Use the binomial theorem to expand each expression. $$ \left(x+y^{2}\right)^{3} $$

Find the probability of the compound event. Tossing a coin three times with the outcomes of three heads

Prove the statement by mathematical induction. $$ 3^{n}>2 n+1, \text { if } n \geq 2 $$

Complete the following for the recursively defined sequence. (a) Find the first four terms. (b) Graph these terms. \(a_{n}=2 a_{n-1}+a_{n-2} ; a_{1}=0, a_{2}=1\)

An ATM access code often consists of a four-digit number. How many codes are possible without giving two accounts the same access code?

Use a formula to find the sum of the arithmetic series. The first 50 terms of the series defined by \(a_{n}=1-3 n\)

Use the binomial theorem to expand each expression. $$ (p-q)^{6} $$

Find the probability of the compound event. Rolling a die three times and obtaining a 5 or 6 on each roll

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

A computer store offers a package in which buyers choose 1 of 2 monitors, 1 of 3 printers, and 1 of 4 types of software. How many different packages can be purchased?

The sum of an arithmetic series with 15 terms is \(255 .\) If \(a_{1}=3,\) find \(a_{15}.\)

Use the binomial theorem to expand each expression. $$ \left(p^{2}-3\right)^{4} $$

Find the probability of the compound event. Rolling a sum of 7 with two dice

Complete the following for the recursively defined sequence. (a) Find the first four terms. (b) Graph these terms. \(a_{n}=\frac{1}{2} a_{n-1}^{3}+1 ; a_{1}=0\)

A red die and a blue die are thrown. How many ways are there for both dice to show an even number?

The sum of an arithmetic series with 20 terms is \(610 .\) If \(a_{20}=59,\) find \(a_{1}.\)

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

Find the probability of the compound event. Rolling a sum of 2 with two dice

Prove the statement by mathematical induction. If \(0

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

How many different 7-digit telephone numbers are possible if the first digit cannot be a 0 or a \(1 ?\)

Use a formula to find the sum of the first 20 terms for the arithmetic sequence. $$ a_{1}=4, d=2 $$

Use the binomial theorem to expand each expression. $$ (3 a-2 b)^{5} $$

Find the probability of the compound event. Rolling a sum other than 7 with two dice

Prove the statement by mathematical induction. \(2^{n}>n^{2}\) for \(n>4\)

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

A menu offers 5 different salads, 10 different entrees, and 4 different desserts. How many ways are there to order a salad, an entrée, and a dessert?

Use a formula to find the sum of the first 20 terms for the arithmetic sequence. $$ a_{1}=-3, d=\frac{2}{3} $$

Use the binomial theorem to expand each expression. $$ \left(1-x^{2}\right)^{4} $$

Find the probability of the compound event. Rolling a die four times without obtaining a 6

Prove the statement by mathematical induction. If \(n \geq 4,\) then \(n !>2^{n},\) where \(n !=n(n-1)(n-2) \cdots(3)(2)(1)\)

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

Evaluate the expression. \(6 !\)

Use a formula to find the sum of the first 20 terms for the arithmetic sequence. $$ a_{1}=10, d=-\frac{1}{2} $$

Use the binomial theorem to expand each expression. $$ \left(2+3 x^{2}\right)^{3} $$

Find the probability of the compound event. Rolling a die four times and obtaining at least one 6

Prove the statement by mathematical induction. \(4^{n}>n^{4}\) for \(n \geq 5\)

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

Evaluate the expression. \(0 !\)

Use a formula to find the sum of the first 20 terms for the arithmetic sequence. $$ a_{1}=0, d=-4 $$

Use the binomial theorem to expand each expression. $$ \left(2 p^{3}-3\right)^{3} $$

Find the probability of the compound event. Drawing four consecutive aces from a standard deck of 52 cards without replacement

Suppose that each of the \(n\) \((n \geq 2)\) people in a room shakes hands with everyone else, but not with himself. Show that the number of handshakes is \(\frac{n^{2}-n}{2}\)