Chapter 12

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

Determine the positive integer values of \(n\), where the geometric sequence \(a_{n}\) satisfies the inequality. $$a_{n}<35, \text { where } a_{n}=2^{n}$$

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

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

Work each problem. Drawing a Card \(\mathrm{A}\) card is drawn from a well-shufflec 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 (count an ace as 1 )

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

Evaluate each expression. Do not use a calculator. $$P(6,0)$$

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

Determine the positive integer values of \(n\), where the geometric sequence \(a_{n}\) satisfies the inequality. $$a_{n}>\frac{1}{10}, \text { where } a_{n}=\frac{1}{2^{n}}$$

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

Work each problem. A woman invites 10 relatives to a party: her mother, 2 uncles, 3 brothers, and 4 cousins. If any one guest is as likely to arrive first as any other, 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.

Use mathematical induction to prove each statement. Assume that \(n\) is a positive integer. $$(1+x)^{n} \geq 1+n x, \text { for } x>-1$$

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

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

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

Determine the positive integer values of \(n\), where the geometric sequence \(a_{n}\) satisfies the inequality.. $$a_{n} \geq \frac{1}{8}, \text { where } a_{n}=8(0.25)^{n}$$

Work each problem. 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.

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

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

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

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

Determine the positive integer values of \(n\), where the geometric sequence \(a_{n}\) satisfies the inequality. .$$a_{n} \geq \frac{2}{27}, \text { where } a_{n}=\frac{6}{3^{n}}$$

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

Work each problem. \(\quad\) Match each probability in parts (a)-(g) with one of the statements in \(\mathrm{A}-\mathrm{F}\). (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 to occur. F. This probability cannot occur.

Determine the positive integer values of \(n\) for which the given statement is not true. $$3^{n}>6 n$$

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

Evaluate each expression. Do not use a calculator. $$C(6,0)$$

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

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

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

Work each problem. Small-Business 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 Joumal analysis of CAl Reports.)

Determine the positive integer values of \(n\) for which the given statement is not true. $$3^{n}>2 n+1$$

Evaluate each expression. Do not use a calculator. $$C(8,1)$$

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

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

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

Work each problem. 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{tabular}{c|c} \hline Organ Transplant & Patients Waiting \\ \hline Heart & \(3,774\) \\ Kidney & \(35,025\) \\ Liver & \(7,920\) \\ Lung & \(2,340\) \end{tabular} Assuming that none of these people needs 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.

Determine the positive integer values of \(n\) for which the given statement is not true. $$2^{n}>n^{2}$$

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

Evaluate each expression. Do not use a calculator. $$C(12,4)$$

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

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

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

Determine the positive integer values of \(n\) for which the given statement is not true. $$n !>2 n$$

Evaluate each expression. Do not use a calculator. \(C(16,3)\)

Write the binomial expansion for each expression. $$(m+n)^{4}$$

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

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

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

Find \(a_{1}\) for each arithmetic sequence. $$a_{5}=27, a_{15}=87$$