91Ó°ÊÓ

Pascal's Triangle Use Pascal's triangle to expand the expression. $$(x-y)^{5}$$

Determine the common ratio, the fifth term, and the \(n\) th term of the geometric sequence. $$2,6,18,54, \dots$$

Which is larger, \((100 !)^{101}\) or (101!) \(^{100}\) ? [ Hint: Try factoring the expressions. Do they have any common factors?]

An architect designs a theater with 15 seats in the first row. 18 in the second, 21 in the third, and so on. If the theater is to have a seating capacity of \(870,\) how many rows must the architect use in his design?

Express the repeating decimal as a fraction. $$0.777 \ldots$$

A culture initially has 5000 bacteria, and its size increases by \(8 \%\) every hour. How many bacteria are present at the end of 5 hours? Find a formula for the number of bacteria present after \(n\) hours.

A ball is dropped from a height of 9 ft. The elasticity of the ball is such that it always bounces up onethird the distance it has fallen. (a) Find the total distance the ball has traveled at the instant it hits the ground the fifth time. (b) Find a formula for the total distance the ball has traveled at the instant it hits the ground the \(n\) th time.

The following is a well-known children's rhyme: As I was going to St. Ives, I met a man with seven wives: Every wife had seven sacks; Every sack had seven cats; Every cat had seven kits; Kits, cats, sacks, and wives, How many were going to St. Ives?