Chapter 12

In Exercises \(1-12\), determine whether the sequence is arithmetic, geometric, or neither. $$2,7,12,17,22, \dots$$

Use mathematical induction to prove that each of the given statements is true for every positive integer \(n .\) $$1+2+2^{2}+2^{3}+2^{4}+\cdots+2^{n-1}=2^{n}-1$$

Evaluate the expression. $$6 !$$

Determine whether the sequence is arithmetic or not. If it is, find the common difference. $$1,3,5,7,9, \dots$$

Find the first five terms of the sequence \(\left\\{a_{n}\right\\}\). $$a_{n}=2 n+6$$

In Exercises \(1-12\), determine whether the sequence is arithmetic, geometric, or neither. $$2,6,18,54,162, \dots$$

Evaluate the expression. $$10 !$$

Determine whether the sequence is arithmetic or not. If it is, find the common difference. $$\frac{1}{3}, \frac{2}{3}, \frac{3}{3}, \frac{4}{3}, \frac{5}{3}, \ldots$$

Find the first five terms of the sequence \(\left\\{a_{n}\right\\}\). $$a_{n}=2^{n}-7$$

In Exercises \(1-12\), determine whether the sequence is arithmetic, geometric, or neither. $$13,13 / 2,13 / 4,13 / 8, \dots$$

Use mathematical induction to prove that each of the given statements is true for every positive integer \(n .\) $$1+3+5+7+\cdots+(2 n-1)=n^{2}$$

Evaluate the expression. $$\frac{8 !}{6 !}$$

Find the first five terms of the sequence \(\left\\{a_{n}\right\\}\). $$a_{n}=\frac{1}{n^{3}}$$

In Exercises \(1-12\), determine whether the sequence is arithmetic, geometric, or neither. $$-1,-\frac{1}{2}, 0, \frac{1}{2}, \dots$$

Use mathematical induction to prove that each of the given statements is true for every positive integer \(n .\) $$1^{2}+2^{2}+3^{2}+\cdots+n^{2}=\frac{n(n+1)(2 n+1)}{6}$$

Evaluate the expression. $$\frac{11 !}{8 !}$$

Determine whether the sequence is arithmetic or not. If it is, find the common difference. $$-9,-6,-3,0, \dots$$

Find the first five terms of the sequence \(\left\\{a_{n}\right\\}\). $$a_{n}=\frac{1}{(n+3)(n+1)}$$

In Exercises \(1-12\), determine whether the sequence is arithmetic, geometric, or neither. $$50,48,46,44, \dots$$

Use mathematical induction to prove that each of the given statements is true for every positive integer \(n .\) $$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots+\frac{1}{2^{n}}=1-\frac{1}{2^{n}}$$

Evaluate the expression. $$\frac{12 !}{9 ! 3 !}$$

Determine whether the sequence is arithmetic or not. If it is, find the common difference. $$\log 1, \log 2, \log 4, \log 8, \log 16, \dots$$

Find the first five terms of the sequence \(\left\\{a_{n}\right\\}\). $$a_{n}=\frac{n}{2^{n}}$$

Use mathematical induction to prove that each of the given statements is true for every positive integer \(n .\) $$\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\dots+\frac{1}{3^{n}}=\frac{1}{2}-\frac{1}{2 \cdot 3^{n}}$$

Evaluate the expression. $$\frac{9 !-8 !}{7 !}$$

Determine whether the sequence is arithmetic or not. If it is, find the common difference. $$\log 3, \log 6, \log 9, \log 12, \log , 15, \dots$$

Find the first five terms of the sequence \(\left\\{a_{n}\right\\}\). $$a_{n}=\sqrt{n^{2}+1}$$

In Exercises \(1-12\), determine whether the sequence is arithmetic, geometric, or neither. $$3,-3 / 2,3 / 4,-3 / 8,3 / 16, \dots$$

Use mathematical induction to prove that each of the given statements is true for every positive integer \(n .\) $$\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\cdots+\frac{1}{n(n+1)}=\frac{n}{n+1}$$

Evaluate the expression. $$\left(\begin{array}{l}6 \\\2\end{array}\right)$$

Determine whether the sequence is arithmetic or not. If it is, find the common difference. $$\frac{1}{3},-\frac{4}{6},-\frac{15}{9},-\frac{32}{12},-\frac{55}{15},-\frac{84}{18}, \dots$$

Find the first five terms of the sequence \(\left\\{a_{n}\right\\}\). $$a_{n}=(-1)^{n} \sqrt{n+2}$$

In Exercises \(1-12\), determine whether the sequence is arithmetic, geometric, or neither. $$-6,-3.7,-1.4, .9,3.2, \ldots$$

Determine whether the sequence is arithmetic or not. If it is, find the common difference. $$\frac{1}{2}, \frac{3}{4}, 1, \frac{3}{4}, \frac{1}{2}, \frac{1}{4}, \dots$$

Find the first five terms of the sequence \(\left\\{a_{n}\right\\}\). $$a_{n}=(-1)^{n+1} n(n-1)$$

In Exercises \(1-12\), determine whether the sequence is arithmetic, geometric, or neither. $$3,3 \sqrt{2}, 6,6 \sqrt{2}, 12,12 \sqrt{2}, \ldots$$

Use mathematical induction to prove that each of the given statements is true for every positive integer \(n .\) $$n+2>n$$

Evaluate the expression. $$\left(\begin{array}{c}100 \\\99\end{array}\right)$$

Write the first five terms of the sequence whose nth term is given. Use them to decide whether the sequence is arithmetic. If it is, list the common difference. $$a_{n}=5+4 n$$

Find the first five terms of the sequence \(\left\\{a_{n}\right\\}\). $$a_{n}=4+(-.1)^{n}$$

In Exercises \(1-12\), determine whether the sequence is arithmetic, geometric, or neither. $$\ln e, \ln e^{2}, \ln e^{3}, \ln e^{4}, \ln e^{5}, \ldots$$

Use mathematical induction to prove that each of the given statements is true for every positive integer \(n .\) $$2 n+2>n$$

Evaluate the expression. $$\left(\begin{array}{c}200 \\\198\end{array}\right)$$

Write the first five terms of the sequence whose nth term is given. Use them to decide whether the sequence is arithmetic. If it is, list the common difference. $$b_{n}=n-\frac{5}{4}$$

Find the first five terms of the sequence \(\left\\{a_{n}\right\\}\). $$a_{n}=5-(.1)^{n}$$

In Exercises \(1-12\), determine whether the sequence is arithmetic, geometric, or neither. $$6,6,6,6,6, \dots$$

Use mathematical induction to prove that each of the given statements is true for every positive integer \(n .\) $$3^{n} \geq 3 n$$

Evaluate the expression. $$\left(\begin{array}{l}4 \\\1\end{array}\right)\left(\begin{array}{l}5 \\\3\end{array}\right)$$

Write the first five terms of the sequence whose nth term is given. Use them to decide whether the sequence is arithmetic. If it is, list the common difference. $$c_{n}=(-1)^{n}$$

Find the first five terms of the sequence \(\left\\{a_{n}\right\\}\). $$a_{n}=(-1)^{n}+3 n$$