91Ó°ÊÓ

Look for a pattern and then write an expression for the general term, or nth term, \(a_{n},\) of each sequence. Answers may vary. $$ 1 \cdot 3,2 \cdot 4,3 \cdot 5,4 \cdot 6, \dots $$

Theaters are often built with more seats per row as the rows move toward the back. The Community Theater has 20 seats in the first row, 22 in the second, 24 in the third, and so on, for 16 rows. How many seats are in the theater?

Look for a pattern and then write an expression for the general term, or nth term, \(a_{n},\) of each sequence. Answers may vary. $$ 0.1,0.01,0.001,0.0001, \ldots $$

Determine whether each infinite geometric series has a limit.If a limit exists, find it. $$3+15+75+\cdots$$

Carrie saves money in an arithmetic sequence: \(\$ 700\) for the first year, another \(\$ 850\) the second, and so on, for 20 years. How much does she save in all (disregarding interest)?

Look for a pattern and then write an expression for the general term, or nth term, \(a_{n},\) of each sequence. Answers may vary. $$ \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \dots $$

It is said that as a young child, the mathematician Karl F. Gauss \((1777-1855)\) was able to compute the sum \(1+2+3+\cdots+100\) very quickly in his head. Explain how Gauss might have done this and present a formula for the sum of the first \(n\) natural numbers. (Hint: \(1+99=100 .)\)

Find the indicated term for each binomial expression. $$\text { Middle, }\left(2 u+3 v^{2}\right)^{10}$$

Determine whether each infinite geometric series has a limit.If a limit exists, find it. $$-6+3-\frac{3}{2}+\frac{3}{4}-\cdots$$

Look for a pattern and then write an expression for the general term, or nth term, \(a_{n},\) of each sequence. Answers may vary. $$ 1,-4,9,-16, \dots $$