91Ó°ÊÓ

the general term of a sequence is given and involves a factorial. Write the first four terms of each sequence. $$ a_{n}=\frac{(n+1) !}{n^{2}} $$

Find the indicated term of the arithmetic sequence with first term, \(a_{1},\) and common difference, \(d .\) Find \(a_{150}\) when \(a_{1}=-60, d=5\)

Use mathematical induction to prove that each statement is true for every positive integer n. \(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots+\frac{1}{2^{n}}=1-\frac{1}{2^{n}}\)

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (y-4)^{4} $$

Fifty people purchase raffle tickets. Three winning tickets are selected at random. If each prize is \(\$ 500,\) in how many different ways can the prizes be awarded?

In Exercises \(17-24,\) write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7},\) the seventh term of the sequence. $$ 1.5,-3,6,-12, \dots $$

the general term of a sequence is given and involves a factorial. Write the first four terms of each sequence. $$ a_{n}=2(n+1) ! $$

Evaluate each expression. \(\frac{7 P_{3}}{3 !}-_{7} C_{3}\)

A fair coin is tossed two times in succession. The sample space of equally likely outcomes is \(\\{H H, H T, T H, T T\\} .\) Find the probability of getting $$\text{two heads.}$$

Find the indicated term of the arithmetic sequence with first term, \(a_{1},\) and common difference, \(d .\) Find \(a_{60}\) when \(a_{1}=35, d=-3\)