91Ó°ÊÓ

Graphing the Terms of a Sequence In Exercises \(27-32,\) use a graphing utility to graph the first 10 terms of the sequence. (Assume that \(n\) begins with \(1 . )\) $$ a_{n}=\frac{2}{3} n $$

Single File In how many orders can four girls and four boys walk through a doorway single file when (a) there are no restrictions? (b) the girls walk through before the boys?

Finding the \(n\) th Term In Exercises \(21 - 30\) , find a formula for \(a _ { n }\) for the arithmetic sequence. $$ a _ { 1 } = - 4 , a _ { 5 } = 16 $$

Graphing the Terms of a Sequence In Exercises \(27-32,\) use a graphing utility to graph the first 10 terms of the sequence. (Assume that \(n\) begins with \(1 .\) . $$ a_{n}=2-\frac{4}{n} $$

Find the probability for the experiment of tossing a six-sided die twice. The sum is \(2,3,\) or 12.

Expanding an Expression In Exercises \(19-40,\) use the Binomial Theorem to expand and simplify the expression. $$(7 a+b)^{3}$$

Proving an Inequality In Exercises \(25-30\) , use mathematical induction to prove the inequality for the indicated integer values of \(n .\) $$ \left(\frac{x}{y}\right)^{n+1}<\left(\frac{x}{y}\right)^{n}, \quad n \geq 1 \text { and } 0 < x < y $$

Writing the \(n\) th Term of a Geometric Sequence, write the first five terms of the geometric sequence. Determine the common ratio and write the \(n\) th term of the sequence as a function of \(n .\) $$ a_{1}=80, \quad a_{k+1}=-\frac{1}{2} a_{k} $$

Finding a Term of a Geometric Sequence, write an expression for the \(n\) th term of the geometric sequence. Then find the indicated term. $$ a_{1}=4, r=\frac{1}{2}, n=10 $$

Expanding an Expression In Exercises \(19-40,\) use the Binomial Theorem to expand and simplify the expression. $$(r+3 s)^{6}$$