91Ó°ÊÓ

Use Pascal’s triangle to expand the expression. $$ \left(x+\frac{1}{x}\right)^{4} $$

Annuity Find the amount of an annuity that consists of 16 quarterly payments of \(\$ 300\) each into an account that pays 8\(\%\) interest per year, compounded quarterly.

The \(n\) th term of a sequence is given. (a) Find the first five terms of the sequence. (b) What is the common ratio \(r ?\) (c) Graph the terms you found in (a). $$ a_{n}=\frac{5}{2}\left(-\frac{1}{2}\right)^{n-1} $$

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

The \(n\) th term of a sequence is given. (a) Find the first five terms of the sequence. (b) What is the common ratio \(r ?\) (c) Graph the terms you found in (a). $$ a_{n}=3^{n-1} $$

\(3-12\) . Find the first four terms and the 100 th term of the sequence. $$ a_{n}=\frac{1}{n^{2}} $$

Use mathematical induction to prove that the formula is true for all natural numbers \(n\). $$ 1 \cdot 3+2 \cdot 4+3 \cdot 5+\cdots+n(n+2)=\frac{n(n+1)(2 n+7)}{6} $$

\(5-8=\) A sequence is given. (a) Find the first five terms of the sequence. (b) What is the common difference \(d ?\) (c) Graph the terms you found in (a). $$ a_{n}=\frac{1}{2}(n-1) $$

Annuity Find the amount of an annuity that consists of 40 annual payments of \(\$ 2000\) each into an account that pays interest of 5\(\%\) per year.

Use mathematical induction to prove that the formula is true for all natural numbers \(n\). $$ 1^{3}+2^{3}+3^{3}+\cdots+n^{3}=\frac{n^{2}(n+1)^{2}}{4} $$