91Ó°ÊÓ

Show that each sequence is arithmetic. Find the common difference, and list the first four terms. $$ \left\\{b_{n}\right\\}=\\{3 n+1\\} $$

Show that each sequence is arithmetic. Find the common difference, and list the first four terms. $$ \left\\{c_{n}\right\\}=\\{6-2 n\\} $$

Evaluate each factorial expression. \(\frac{9 !}{6 !}\)

Show that each sequence is geometric. Then find the common ratio and list the first four terms. $$ \left\\{a_{n}\right\\}=\left\\{-3\left(\frac{1}{2}\right)^{n}\right\\} $$

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ \frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\cdots+\frac{1}{n(n+1)} \equiv \frac{n}{n+1} $$

Show that each sequence is arithmetic. Find the common difference, and list the first four terms. $$ \left\\{a_{n}\right\\}=\\{4-2 n\\} $$

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ \frac{1}{1 \cdot 3}+\frac{1}{3 \cdot 5}+\frac{1}{5 \cdot 7}+\cdots+\frac{1}{(2 n-1)(2 n+1)}=\frac{n}{2 n+1} $$

Evaluate each factorial expression. \(\frac{12 !}{10 !}\)

Show that each sequence is geometric. Then find the common ratio and list the first four terms. $$ \left\\{c_{n}\right\\}=\left\\{\frac{2^{n-1}}{4}\right\\} $$

Show that each sequence is arithmetic. Find the common difference, and list the first four terms. $$ \left\\{t_{n}\right\\}=\left\\{\frac{1}{2}-\frac{1}{3} n\right\\} $$