91Ó°ÊÓ

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

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. $$c_{n}=(-1)^{n}$$

In Exercises \(13-22,\) one term and the common ratio r of a geometric sequence are given. Find the sixth term and a formula for the nth term. $$a_{1}=4, r=\frac{1}{4}$$

In Exercises \(13-22,\) one term and the common ratio r of a geometric sequence are given. Find the sixth term and a formula for the nth term. $$a_{3}=1 / 2, r=3$$

In Exercises \(39-42,\) find the kth partial sum of the geometric sequence \(\left\\{a_{n}\right\\}\) with common ratio \(r\). $$k=7, a_{2}=6, r=2$$

Find the kth partial sum of the arithmetic sequence \(\left\\{a_{n}\right\\}\) with common difference d. $$k=9, a_{1}=6, a_{9}=-24$$

A ball is dropped from a height of 8 feet. On each bounce, it rises to half its previous height. When the ball hits the ground for the seventh time, how far has it traveled?

Starting with your parents, how many ancestors do you have for the preceding ten generations?

A car that sold for \(\$ 8000\) depreciates in value \(25 \%\) each year. What is it worth after five years?

Find the sum of all the integer multiples of 7 from 7 to 700 .