Chapter 12

CHECKING ANALYTIC SKILLS Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. Do not use a calculator. $$a_{3}=-2, r=4$$

Write the first five terms of each sequence. Do not use a calculator. $$a_{n}=(-2)^{n}(n)$$

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

Write the first five terms of each arithmetic sequence. Do not use a calculator. $$a_{1}=5, d=-2$$

Evaluate each expression. Do not use a calculator. $$3 ! \cdot 4$$

CHECKING ANALYTIC SKILLS Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. Do not use a calculator. $$a_{4}=243, r=-3$$

Write the first five terms of each sequence. Do not use a calculator. $$a_{n}=(-1)^{n}(2 n)$$

Use mathematical induction to prove each statement. Assume that \(n\) is a positive integer. $$1^{2}+2^{2}+3^{2}+\cdots+n^{2}=\frac{n(n+1)(2 n+1)}{6}$$

Write the first five terms of each arithmetic sequence. Do not use a calculator. $$a_{1}=4, d=3$$

Evaluate each expression. Do not use a calculator. $$4 ! \cdot 5$$

Evaluate the following. In Exercises 17 and 18 , express the answer in terms of n. Do not use a calculator. $$\left(\begin{array}{c}12 \\\12\end{array}\right)$$

CHECKING ANALYTIC SKILLS Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. Do not use a calculator. $$a_{4}=18, r=2$$

Write the first five terms of each sequence. Do not use a calculator. $$a_{n}=(-1)^{n-1}(n+1)$$

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

Write each event in set notation. Give the probability of the event. A student gives the answer to a probability problem as \(\frac{6}{5}\). Explain why this answer must be incorrect.

Evaluate the following. In Exercises 17 and 18 , express the answer in terms of n. Do not use a calculator. $$\left(\begin{array}{l}8 \\\3\end{array}\right)$$

CHECKING ANALYTIC SKILLS Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. Do not use a calculator. $$-4,-12,-36,-108, \dots$$

Write the first five terms of each sequence. Do not use a calculator. $$a_{n}=\frac{4 n-1}{n^{2}+2}$$

Use mathematical induction to prove each statement. Assume that \(n\) is a positive integer. $$5 \cdot 6+5 \cdot 6^{2}+5 \cdot 6^{3}+\cdots+5 \cdot 6^{n}=6\left(6^{n}-1\right)$$

If the probability of an event is 0.857 what is the probability that the event will not occur?

Evaluate each expression. Do not use a calculator. If \(n\) is a positive integer greater than \(1,\) is \((n-1) ! \cdot n\) always equal to \(n ! ?\)

Evaluate the following. In Exercises 17 and 18 , express the answer in terms of n. Do not use a calculator. $$\left(\begin{array}{l}9 \\\7\end{array}\right)$$

CHECKING ANALYTIC SKILLS Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. Do not use a calculator. $$-2,6,-18,54, \dots$$

Write the first five terms of each sequence. Do not use a calculator. $$a_{n}=\frac{n^{2}-1}{n^{2}+1}$$

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

Work each problem. Drawing a Marble A marble is drawn at random from a box containing 3 yellow, 4 white, and 8 blue marbles. Find the probabilities in parts (a)-(c). A. A yellow marble is drawn. B. A black marble is drawn. C. The marble is yellow or white. D. What are the odds in favor of drawing a yellow marble? E. What are the odds against drawing a blue marble?

Find \(a_{8}\) and \(a_{n}\) for each arithmetic sequence. $$a_{1}=5, d=2$$

Evaluate each expression. Do not use a calculator. $$P(7,7)$$

CHECKING ANALYTIC SKILLS Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. Do not use a calculator. $$\frac{4}{5}, 2,5, \frac{25}{2}, \dots$$

Your friend does not understand what is meant by the \(n\)th term, or general term, of a sequence. How would you explain this idea?

Use mathematical induction to prove each statement. Assume that \(n\) is a positive integer. $$7 \cdot 8+7 \cdot 8^{2}+7 \cdot 8^{3}+\dots+7 \cdot 8^{n}=8\left(8^{n}-1\right)$$

Work each problem. \(\quad\) A baseball player with a batting average of .300 comes to bat. What are the odds in favor of his getting a hit?

Find \(a_{8}\) and \(a_{n}\) for each arithmetic sequence. $$a_{1}=-3, d=-4$$

Evaluate each expression. Do not use a calculator. $$P(5,5)$$

Evaluate the following. In Exercises 17 and 18 , express the answer in terms of n. Do not use a calculator. $$\left(\begin{array}{c}20 \\\15\end{array}\right)$$

CHECKING ANALYTIC SKILLS Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. Do not use a calculator. $$\frac{1}{2}, \frac{2}{3}, \frac{8}{9}, \frac{32}{27}, \ldots$$

Use mathematical induction to prove each 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 \(a_{8}\) and \(a_{n}\) for each arithmetic sequence. $$a_{3}=2, d=1$$

Evaluate each expression. Do not use a calculator. $$P(9,2)$$

Evaluate the following. In Exercises 17 and 18 , express the answer in terms of n. Do not use a calculator. $$\left(\begin{array}{l}5 \\\0\end{array}\right)$$

CHECKING ANALYTIC SKILLS Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. Do not use a calculator. $$10,-5, \frac{5}{2},-\frac{5}{4}, \dots$$

Decide whether each sequence is finite or infinite. The sequence of days of the week

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

Work each problem. If the odds that it will rain are 4 to \(5,\) what is the probability of rain?

Find \(a_{8}\) and \(a_{n}\) for each arithmetic sequence. $$a_{4}=5, d=-2$$

Evaluate each expression. Do not use a calculator. $$P(10,3)$$

CHECKING ANALYTIC SKILLS Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. Do not use a calculator. $$3,-\frac{9}{4}, \frac{27}{16},-\frac{81}{64}, \dots$$

Decide whether each sequence is finite or infinite. The sequence of dates in the month of November

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

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