Chapter 10

Show that the given sequence is geometric, and find the common ratio. $$5,-\frac{5}{4}, \frac{5}{16}, \ldots, 5\left(-\frac{1}{4}\right)^{n-1}, \ldots$$

Exer. 1-8: Find the number. $$ C(7,3) $$

\(P(7,3)\)

Exer. 1-26: Prove that the statement is true for every positive integer \(n\). $$ 2+4+6+\cdots+2 n=n(n+1) $$

Show that the given sequence is geometric, and find the common ratio. $$\frac{1}{7}, \frac{3}{7}, \frac{9}{7}, \ldots, \frac{1}{7}(3)^{n-1}, \ldots$$

Exer. 1-8: Find the number. $$ C(8,4) $$

\(P(8,5)\)

Exer. 1-26: Prove that the statement is true for every positive integer \(n\). $$ 1+4+7+\cdots+(3 n-2)=\frac{n(3 n-1)}{2} $$

Exer. 1-2: Show that the given sequence is arithmetic, and find the common difference. $$ 53,48,43, \ldots, 58-5 n, \ldots $$

Find the \(n\)th term, the fifth term, and the eighth term of the geometric sequence. $$8,4,2,1, \ldots$$

Exer. 1-8: Find the number. $$ C(9,8) $$

\(P(9,6)\)

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

Exer. 3-10: Find the \(n\)th term, the fifth term, and the tenth term of the arithmetic sequence. $$ 2,6,10,14, \ldots $$

Find the \(n\)th term, the fifth term, and the eighth term of the geometric sequence. $$4,1.2,0.36,0.108, \ldots$$

(a) an even number (b) a number divisible by 5 (c) an even number or a number divisible by 5

Exer. 1-8: Find the number. $$ C(6,2) $$

\( P(5,3)\)

Evaluate the expression. $$ 5 ! 0 ! $$

Exer. 1-26: Prove that the statement is true for every positive integer \(n\). $$ 3+9+15+\cdots+(6 n-3)=3 n^{2} $$

Exer. 3-10: Find the \(n\)th term, the fifth term, and the tenth term of the arithmetic sequence. $$ 16,13,10,7, \ldots $$

Find the \(n\)th term, the fifth term, and the eighth term of the geometric sequence. $$300,-30,3,-0.3, \ldots$$

\(P(5,5)\)

Evaluate the expression. $$ \frac{8 !}{5 !} $$

Exer. 1-26: Prove that the statement is true for every positive integer \(n\). $$ 2+7+12+\cdots+(5 n-3)=\frac{1}{2} n(5 n-1) $$

Exer. 3-10: Find the \(n\)th term, the fifth term, and the tenth term of the arithmetic sequence. $$ 3,2.7,2.4,2.1, \ldots $$

Find the \(n\)th term, the fifth term, and the eighth term of the geometric sequence. $$1,-\sqrt{3}, 3,-3 \sqrt{3}, \ldots$$

\( P(4,4)\)

Evaluate the expression. $$ \frac{6 !}{3 !} $$

Exer. 1-26: Prove that the statement is true for every positive integer \(n\). $$ 2+6+18+\cdots+2 \cdot 3^{n-1}=3^{n}-1 $$

Exer. 3-10: Find the \(n\)th term, the fifth term, and the tenth term of the arithmetic sequence. $$ -6,-4.5,-3,-1.5, \ldots $$

Find the \(n\)th term, the fifth term, and the eighth term of the geometric sequence. $$5,25,125,625, \ldots$$

Exer. 1-8: Find the number. $$ C(7,0) $$

\(P(6,1)\)

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

Exer. 3-10: Find the \(n\)th term, the fifth term, and the tenth term of the arithmetic sequence. $$ -7,-3.9,-0.8,2.3, \ldots $$

Find the \(n\)th term, the fifth term, and the eighth term of the geometric sequence. $$2,6,18,54, \ldots$$

Exer. 1-8: Find the number. $$ C(5,5) $$

\(P(5,1)\)

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

Exer. 3-10: Find the \(n\)th term, the fifth term, and the tenth term of the arithmetic sequence. $$ x-8, x-3, x+2, x+7, \ldots $$

Find the \(n\)th term, the fifth term, and the eighth term of the geometric sequence. $$4,-6,9,-13.5, \ldots$$

Exer. 9-10: Find the number of possible color arrangements for the 12 given disks, arranged in a row. 5 black, 3 red, 2 white, 2 green

Exer. 1-26: Prove that the statement is true for every positive integer \(n\). $$ 1^{2}+2^{2}+3^{2}+\cdots+n^{2}=\frac{n(n+1)(2 n+1)}{6} $$

Exer. 3-10: Find the \(n\)th term, the fifth term, and the tenth term of the arithmetic sequence. $$ \ln 3, \ln 9, \ln 27, \ln 81, \ldots $$

Find the \(n\)th term, the fifth term, and the eighth term of the geometric sequence. $$162,-54,18,-6, \ldots$$

Exer. 9-10: Find the number of possible color arrangements for the 12 given disks, arranged in a row. 3 black, 3 red, 3 white, 3 green

Evaluate the expression. $$ \left(\begin{array}{l} 8 \\ 4 \end{array}\right) $$

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

Exer. 3-10: Find the \(n\)th term, the fifth term, and the tenth term of the arithmetic sequence. $$ \log 1000, \log 100, \log 10, \log 1, \ldots $$