91Ó°ÊÓ

In Exercises \(31-38,\) use the given information about the geometric sequence \(\left\\{a_{n}\right\\}\) to find as and a formula for \(a_{n}\). $$a_{1}=1 / 6, a_{2}=-1 / 18$$

Expand and (where possible) simplify the expression. $$\left(\sqrt{c}+\frac{1}{\sqrt{c}}\right)^{7}$$

Use the Extended Principle of Mathematical Induction (Exercise 28 ) to prove the given statement. $$2^{n}>n^{2} \quad \text { for all } n \geq 5$$

The first term \(a_{1}\) and the common difference d of an arithmetic sequence are given. Find the fifth term and the formula for the nth term. $$a_{1}=-.1, d=-8$$

Find the first five terms of the recursively defined sequence. $$a_{0}=2, a_{1}=3, \text { and } a_{n}=\left(a_{n-1}\right)\left(\frac{1}{2} a_{n-2}\right) \quad \text { for } n \geq 2$$

In Exercises \(31-38,\) use the given information about the geometric sequence \(\left\\{a_{n}\right\\}\) to find as and a formula for \(a_{n}\). $$a_{1}=1 / 2, a_{2}=5$$

Expand and (where possible) simplify the expression. $$\left(x^{-3}+x\right)^{4}$$

Use the given information about the arithmetic sequence with common difference d to find a and a formula for \(a_{n}\). $$a_{4}=12, d=2$$

Expand and (where possible) simplify the expression. $$\left(3 x^{-2}-x^{2}\right)^{6}$$

In Exercises \(31-38,\) use the given information about the geometric sequence \(\left\\{a_{n}\right\\}\) to find as and a formula for \(a_{n}\). $$a_{1}=\sqrt{7}, a_{2}=\sqrt{42}$$