Chapter 10

Find the \(n\)th term, the fifth term, and the eighth term of the geometric sequence. $$1,-x^{2}, x^{4},-x^{6}, \ldots$$

If three coins are flipped, find the probability that exactly two heads turn up.

Find the number of distinguishable permutations of the letters in the word bookkeeper.

Evaluate the expression. $$ \left(\begin{array}{c} 13 \\ 4 \end{array}\right) $$

Exer. 1-26: Prove that the statement is true for every positive integer \(n\). $$ \frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\cdots+\frac{1}{n(n+1)}=\frac{n}{n+1} $$

Exer. 11-12: Find the common difference for the arithmetic sequence with the specified terms. $$ a_{2}=21, a_{6}=-11 $$

Find the \(n\)th term, the fifth term, and the eighth term of the geometric sequence. $$1,-\frac{x}{3}, \frac{x^{2}}{9},-\frac{x^{3}}{27}, \ldots$$

If four coins are flipped, find the probability of obtaining two heads and two tails.

Find the number of distinguishable permutations of the letters in the word moon. List all the permutations.

Evaluate the expression. $$ \left(\begin{array}{c} 52 \\ 2 \end{array}\right) $$

Exer. 1-26: Prove that the statement is true for every positive integer \(n\). $$ \begin{aligned} \frac{1}{1 \cdot 2 \cdot 3}+\frac{1}{2 \cdot 3 \cdot 4} &+\frac{1}{3 \cdot 4 \cdot 5}+\cdots+\\\ & \frac{1}{n(n+1)(n+2)}=\frac{n(n+3)}{4(n+1)(n+2)} \end{aligned} $$

Exer. 11-12: Find the common difference for the arithmetic sequence with the specified terms. $$ a_{4}=14, a_{11}=35 $$

Find the \(n\)th term, the fifth term, and the eighth term of the geometric sequence. $$2,2^{x+1}, 2^{2 x+1}, 2^{3 x+1}, \ldots$$

If \(P(E)=\frac{5}{7}\), find \(O(E)\) and \(O\left(E^{\prime}\right)\)

Ten people wish to play in a basketball game. In how many different ways can two teams of five players each be formed?

Rewrite as an expression that does not contain factorials. $$ \frac{n !}{(n-2) !} $$

How many three-digit numbers can be formed from the digits \(1,2,3,4\), and 5 if repetitions (a) are not allowed? (b) are allowed?

Exer. 1-26: Prove that the statement is true for every positive integer \(n\). $$ 3+3^{2}+3^{3}+\cdots+3^{n}=\frac{3}{2}\left(3^{n}-1\right) $$

Exer. 13-18: Find the specified term of the arithmetic sequence that has the two given terms. $$ a_{12} ; \quad a_{1}=9.1, \quad a_{2}=7.5 $$

If \(P(E)=0.4\), find \(O(E)\) and \(O\left(E^{\prime}\right)\)

A student may answer any six of ten questions on an examination. (a) In how many ways can six questions be selected? (b) How many selections are possible if the first two questions must be answered?

Rewrite as an expression that does not contain factorials. $$ \frac{(n+1) !}{(n-1) !} $$

Exer. 1-26: Prove that the statement is true for every positive integer \(n\). $$ 1^{3}+3^{3}+5^{3}+\cdots+(2 n-1)^{3}=n^{2}\left(2 n^{2}-1\right) $$

Exer. 13-18: Find the specified term of the arithmetic sequence that has the two given terms. $$ a_{11} ; \quad a_{1}=2+\sqrt{2}, \quad a_{2}=3 $$

Find all possible values of \(r\) for a geometric sequence with the two given terms. $$a_{4}=3, a_{6}=9$$

Exer. 15-16: Consider any eight points such that no three are collinear. How many lines are determined?

Rewrite as an expression that does not contain factorials. $$ \frac{(2 n+2) !}{(2 n) !} $$

How many numbers can be formed from the digits \(1,2,3\), and 4 if repetitions are not allowed? (Note: 42 and 231 are examples of such numbers.)

Exer. 1-26: Prove that the statement is true for every positive integer \(n\). $$ n<2^{n} $$

Exer. 13-18: Find the specified term of the arithmetic sequence that has the two given terms. $$ a_{1} ; \quad a_{6}=2.7, \quad a_{7}=5.2 $$

Find all possible values of \(r\) for a geometric sequence with the two given terms. $$a_{3}=4, a_{7}=\frac{1}{4}$$

If \(O\left(E^{\prime}\right)\) are 7 to 3 , find \(O(E)\) and \(P(E)\).

Exer. 15-16: Consider any eight points such that no three are collinear. How many triangles are determined?

Rewrite as an expression that does not contain factorials. $$ \frac{(3 n+1) !}{(3 n-1) !} $$

Determine the number of positive integers less than 10,000 that can be formed from the digits \(1,2,3\), and 4 if repetitions are allowed.

Exer. 1-26: Prove that the statement is true for every positive integer \(n\). $$ 1+2 n \leq 3^{n} $$

Exer. 13-18: Find the specified term of the arithmetic sequence that has the two given terms. $$ a_{1} ; \quad a_{8}=47, \quad a_{9}=53 $$

Book arrangement A student has five mathematics books, four history books, and eight fiction books. In how many different ways can they be arranged on a shelf if books in the same category are kept next to one another?

Use the binomial theorem to expand and simplify. $$ (4 x-y)^{3} $$

Basketball standings If eight basketball teams are in a tournament, find the number of different ways that first, second, and third place can be decided, assuming ties are not allowed.

Exer. 1-26: Prove that the statement is true for every positive integer \(n\). $$ 1+2+3+\cdots+n<\frac{1}{8}(2 n+1)^{2} $$

Exer. 13-18: Find the specified term of the arithmetic sequence that has the two given terms. $$ a_{15} ; \quad a_{3}=7, \quad a_{20}=43 $$

Find the seventh term of the geometric sequence whose second and third terms are 2 and \(-\sqrt{2}\).

A basketball squad consists of twelve players. (a) Disregarding positions, in how many ways can a team of five be selected? (b) If the center of a team must be selected from two specific individuals on the squad and the other four members of the team from the remaining ten players, find the number of different teams possible.

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

Exer. 1-26: Prove that the statement is true for every positive integer \(n\). $$ \text { If } 0

Exer. 13-18: Find the specified term of the arithmetic sequence that has the two given terms. $$ a_{10} ; \quad a_{2}=1, \quad a_{18}=49 $$

Given a geometric sequence with \(a_{4}=4\) and \(a_{7}=12\), find \(r\) and \(a_{10}\) .

A football squad consists of three centers, ten linemen who can play either guard or tackle, three quarterbacks, six halfbacks, four ends, and four fullbacks. A team must have one center, two guards, two tackles, two ends, two halfbacks, a quarterback, and a fullback. In how many different ways can a team be selected from the squad?

Use the binomial theorem to expand and simplify. $$ (x+y)^{6} $$