91Ó°ÊÓ

You are dealt one card from a standard 52-card deck. Find the probability of being dealt $$\text{a queen.}$$

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. $$ 3,12,48,192, \dots $$

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ \left(x^{2}+2 y\right)^{4} $$

are defined using recursion formulas. Write the first four terms of each sequence. $$ a_{1}=4 \text { and } a_{n}=2 a_{n-1}+3 \text { for } n \geq 2 $$

Use mathematical induction to prove that each statement is true for every positive integer n. \(1+3+3^{2}+\cdots+3^{n-1}=\frac{3^{n}-1}{2}\)

are defined using recursion formulas. Write the first four terms of each sequence. $$ a_{1}=5 \text { and } a_{n}=3 a_{n-1}-1 \text { for } n \geq 2 $$

You are dealt one card from a standard 52-card deck. Find the probability of being dealt $$\text{a diamond.}$$

Fifty people purchase raffle tickets. Three winning tickets are selected at random. If first prize is \(\$ 1000,\) second prize is \(\$ 500,\) and third prize is \(\$ 100,\) in how many different ways can the prizes be awarded?

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ \left(x^{2}+y\right)^{4} $$

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. $$ 3,15,75,375, \dots $$