Chapter 8

Find \(a_{8}\) and \(a_{n}\) for each arithmetic sequence. $$a_{1}=8, a_{2}=6$$

Evaluate the following. In Exercises 17 and \(18,\) express the answer in terms of \(n .\) Do not use a calculator. $$\left(\begin{array}{c}n \\\n-1\end{array}\right)$$

Find \(a_{1}\) and \(r\) for each geometric sequence. $$a_{3}=5, a_{8}=\frac{1}{625}$$

If the odds that a candidate will win an election are 3 to \(2,\) what is the probability that the candidate will lose?

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

Evaluate each expression. $$P(5,1)$$

Decide whether each sequence is finite or infinite. $$1,2,3,4, \dots$$

Find \(a_{8}\) and \(a_{n}\) for each arithmetic sequence. $$a_{1}=6, a_{2}=3$$

Evaluate the following. In Exercises 17 and \(18,\) express the answer in terms of \(n .\) Do not use a calculator. $$\left(\begin{array}{c}n \\\n-2\end{array}\right)$$

Find \(a_{1}\) and \(r\) for each geometric sequence. $$a_{2}=-6, a_{7}=-192$$

Drawing a Card A card is drawn from a well-shuffled deck of 52 cards. Find the probability that the card is as follows. (a) a 9 (b) black (c) a black 9 (d) a heart (e) a face card (K, Q, or J of any suit) (f) red or a 3 (g) less than a 4 (consider aces as 1 s)

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

Evaluate each expression. $$P(6,0)$$

Decide whether each sequence is finite or infinite. $$-1,-2,-3,-4, \dots$$

Find \(a_{8}\) and \(a_{n}\) for each arithmetic sequence. $$a_{10}=6, a_{12}=15$$

Find \(a_{1}\) and \(r\) for each geometric sequence. $$a_{4}=-\frac{1}{4}, a_{9}=-\frac{1}{128}$$

How many terms are there in the expansion of \((x+y)^{8} ?\)

Mrs. Schmulen invites 10 relatives to a party: her mother, 2 uncles, 3 brothers, and 4 cousins. If the chances of any one guest arriving first are equally likely, find each probability. (a) The first guest is an uncle or a brother. (b) The first guest is a brother or a cousin. (c) The first guest is a brother or her mother.

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

Evaluate each expression. $$C(4,2)$$

Decide whether each sequence is finite or infinite. $$a_{1}=3 ; \text { for } 2 \leq n \leq 10, a_{n}=3 \cdot a_{n-1}$$

Find \(a_{8}\) and \(a_{n}\) for each arithmetic sequence. $$a_{15}=8, a_{17}=2$$

How many terms are there in the expansion of \((x+y)^{10} ?\)

Find \(a_{1}\) and \(r\) for each geometric sequence. $$a_{3}=50, a_{7}=0.005$$

Two dice are rolled. Find the probability of each event. (a) The sum is at least \(10 .\) (b) The sum is either 7 or at least \(10 .\) (c) The sum is 2 or the dice both show the same number.

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

Evaluate each expression. $$C(9,3)$$

Decide whether each sequence is finite or infinite. $$a_{1}=1 ; a_{2}=3 ; \text { for } n \geq 3, a_{n}=a_{n-1}+a_{n-2}$$

Find \(a_{8}\) and \(a_{n}\) for each arithmetic sequence. $$a_{1}=x, a_{2}=x+3$$

What are the first and last terms in the expansion of \((2 x+3 y)^{4} ?\)

Use the formula for \(S_{n}\) to find the sum of the first five terms for each geometric sequence. Round the answers for Exercises 25 and 26 to the nearest hundredth. $$2,8,32,128, \dots$$

Concept Check \(\quad\) Match each probability in parts (a)-(g) with one of the statements in A-F. (a) \(P(E)=-0.1\) (b) \(P(E)=0.01\) (c) \(P(E)=1\) (d) \(P(E)=2\) (e) \(P(E)=0.99\) (f) \(P(E)=0\) (g) \(P(E)=0.5\) A. The event is certain to occur. B. The event is impossible. C. The event is very likely to occur. D. The event is very unlikely to occur. E. The event is just as likely to occur as not occur. F. This probability cannot occur.

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

Evaluate each expression. $$C(6,0)$$

Find the first four terms of each sequence. $$a_{1}=-2, a_{n}=a_{n-1}+3, \text { for } n>1$$

Use the formula for \(S_{n}\) to find the sum of the first five terms for each geometric sequence. Round the answers for Exercises 25 and 26 to the nearest hundredth. $$4,16,64,256, \dots$$

Describe in your own words how you would determine the binomial coefficient for the fifth term in the expansion of \((x+y)^{8}\)

Loan The probability that a bank with assets greater than or equal to \(\$ 30\) billion will make a loan to a small business is \(0.002 .\) What are the odds against such a bank making a small-business loan? (Source: The Wall Street Journal analysis of CAI Reports.)

Find all positive integers n for which the given statement is not true. $$n !>2 n$$

Evaluate each expression. $$C(8,1)$$

Find the first four terms of each sequence. $$a_{1}=-1, a_{n}=a_{n-1}-4, \text { for } n>1$$

Find \(a_{8}\) and \(a_{n}\) for each arithmetic sequence. $$a_{3}=\pi+2 \sqrt{e}, a_{4}=\pi+3 \sqrt{e}$$

Use the formula for \(S_{n}\) to find the sum of the first five terms for each geometric sequence. Round the answers for Exercises 25 and 26 to the nearest hundredth. $$18,-9, \frac{9}{2},-\frac{9}{4}, \dots$$

Write the binomial expansion for each expression. $$(x+y)^{6}$$

In a recent year there were \(51,277\) people waiting for an organ transplant. The following table lists the number of patients waiting for the most common types of transplants. $$\begin{array}{l|c} \hline \text { Organ Transplant } & \text { Patients Waiting } \\ \hline \text { Heart } & 3,774 \\ \text { Kidney } & 35,025 \\ \text { Liver } & 7,920 \\ \text { Lung } & 2,340 \end{array}$$ Assuming that none of these people need two or more transplants, approximate the probability that a transplant patient chosen at random will need (a) a kidney or a heart. (b) neither a kidney nor a heart.

Prove each statement by mathematical induction. \(\left(a^{m}\right)^{n}=a^{m n}\) (Assume that \(a\) and \(m\) are constant.)

Evaluate each expression. $$C(12,4)$$

Find the first four terms of each sequence. $$a_{1}=1, a_{2}=1, a_{n}=a_{n-1}+a_{n-2}, \text { for } n \geq 3$$ (the Fibonacci sequence)

Find \(a_{8}\) and \(a_{n}\) for each arithmetic sequence. $$a_{4}=e+2 \sqrt{\pi}, a_{5}=e+3 \sqrt{\pi}$$

Use the formula for \(S_{n}\) to find the sum of the first five terms for each geometric sequence. Round the answers for Exercises 25 and 26 to the nearest hundredth. $$12,-4, \frac{4}{3},-\frac{4}{9}, \dots$$