Chapter 13

\(9-12\) . Find the \(n\) th term of the arithmetic sequence with given first term and common difference \(d\) What is the 10 the term? $$ a=\frac{5}{2}, d=-\frac{1}{2} $$

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

Use Pascal’s triangle to expand the expression. $$ \left(x^{2} y-1\right)^{5} $$

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

Annuity What is the present value of an annuity that consists of 20 semiannual payments of \(\$ 1000\) at an interest rate of 9\(\%\) per year, compounded semiannully?

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

Use Pascal’s triangle to expand the expression. $$ (1+\sqrt{2})^{6} $$

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

Annuity What is the present value of an annuity that consists of 30 monthly payments of \(\$ 300\) at an interest rate of 8\(\%\) per year, compounded monthly.

\(13-20\) . Determine whether the sequence is arithmetic. If is arithmetic, find the common difference. $$ 5,8,11,14, \dots $$

\(13-18\) Find the first five terms of the given recursively defined sequence. $$ a_{n}=2\left(a_{n-1}-2\right) \quad \text { and } \quad a_{1}=3 $$

Use Pascal’s triangle to expand the expression. $$ (2 x-3 y)^{3} $$

Determine whether the sequence is geometric. If is geometric, find the common ratio. $$ 2,4,8,16, \dots $$

Funding an Annuity How much money must be invested now at 9\(\%\) per year, compounded semiannully, to fund an annuity of 20 payments of \(\$ 200\) each, paid every 6 months, the first payment being 6 months from now?

\(13-20\) . Determine whether the sequence is arithmetic. If is arithmetic, find the common difference. $$ 3,6,9,13, \dots $$

\(13-18\) Find the first five terms of the given recursively defined sequence. $$ a_{n}=\frac{a_{n-1}}{2} \quad \text { and } \quad a_{1}=-8 $$

Use Pascal’s triangle to expand the expression. $$ \left(1+x^{3}\right)^{3} $$

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

Determine whether the sequence is geometric. If is geometric, find the common ratio. $$ 2,6,18,36, \dots $$

Funding an Annuity How much money must be invested now at 9\(\%\) per year, compounded semiannully, to fund an annuity of 20 payments of \(\$ 200\) each, paid every 6 months, the first payment being 6 months from now?

\(13-20\) . Determine whether the sequence is arithmetic. If is arithmetic, find the common difference. $$ 2,4,8,16, \dots $$

\(13-18\) Find the first five terms of the given recursively defined sequence. $$ a_{n}=2 a_{n-1}+1 \quad \text { and } \quad a_{1}=1 $$

Use Pascal’s triangle to expand the expression. $$ \left(\frac{1}{x}-\sqrt{x}\right)^{5} $$

Show that \(n^{2}+n\) is divisible by 2 for all natural numbers \(n\)

Determine whether the sequence is geometric. If is geometric, find the common ratio. $$ 3, \frac{3}{2}, \frac{3}{4}, \frac{3}{8}, \dots $$

Financing a Car A woman wants to borrow \(\$ 12,000\) to buy a car. She wants to repay the loan by monthly installments for 4 years. If the interest rate on this loan is 10\(\frac{1}{2} \%\) per year, compounded monthly, what is the amount of each payment?

\(13-20\) . Determine whether the sequence is arithmetic. If is arithmetic, find the common difference. $$ 2,4,6,8, \dots $$

\(13-18\) Find the first five terms of the given recursively defined sequence. $$ a_{n}=\frac{1}{1+a_{n-1}} \quad \text { and } \quad a_{1}=1 $$

Use Pascal’s triangle to expand the expression. $$ \left(2+\frac{x}{2}\right)^{5} $$

Show that \(5^{n}-1\) is divisible by 4 for all natural numbers \(n\)

Determine whether the sequence is geometric. If is geometric, find the common ratio. $$ 27,-9,3,-1, \ldots $$

\(13-20\) . Determine whether the sequence is arithmetic. If is arithmetic, find the common difference. $$ 3, \frac{3}{2}, 0,-\frac{3}{2}, \dots $$

\(13-18\) Find the first five terms of the given recursively defined sequence. $$ a_{n}=a_{n-1}+a_{n-2} \quad \text { and } \quad a_{1}=1, a_{2}=2 $$

Show that \(n^{2}-n+41\) is odd for all natural numbers \(n\)

Determine whether the sequence is geometric. If is geometric, find the common ratio. $$ \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \dots $$

Mortgage What is the monthly payment on a 30 -year mortgage of \(\$ 100,000\) at 8\(\%\) interest per year, compounded monthly? What is the total amount paid on this loan over the 30 -year period?

\(13-20\) . Determine whether the sequence is arithmetic. If is arithmetic, find the common difference. $$ \ln 2, \ln 4, \ln 8, \ln 16, \ldots $$

Determine whether the sequence is geometric. If is geometric, find the common ratio. $$ e^{2}, e^{4}, e^{6}, e^{8}, \dots $$

\(13-18\) Find the first five terms of the given recursively defined sequence. $$ a_{n}=a_{n-1}+a_{n-2}+a_{n-3} \quad \text { and } \quad a_{1}=a_{2}=a_{3}=1 $$

Evaluate the expression. $$ \left(\begin{array}{l}{8} \\ {3}\end{array}\right) $$

Show that \(n^{3}-n+3\) is divisible by 3 for all natural numbers \(n\)

Mortgage What is the monthly payment on a 15 -year mortgage of \(\$ 200,000\) at 6\(\%\) interest? What is the total amount paid on this loan over the 15 -year period?

Determine whether the sequence is arithmetic. If it is arithmetic, find the common difference. $$ 2.6,4.3,6.0,7.7, \ldots $$

Determine whether the sequence is geometric. If is geometric, find the common ratio. $$ 1.0,1.1,1.21,1.331, \dots $$

\(19-24\) . Use a graphing calculator to do the following. (a) Find the first 10 terms of the sequence. (b) Graph the first 10 terms of the sequence. $$ a_{n}=4 n+3 $$

Evaluate the expression. $$ \left(\begin{array}{c}{100} \\ {98}\end{array}\right) $$

Show that \(8^{n}-3^{n}\) is divisible by 5 for all natural numbers \(n\)

Mortgage Dr. Gupta is considering a 30 -year mortgage at 6\(\%\) interest. She can make payments of \(\$ 3500\) a month. What size loan can she afford?

\(13-20\) . Determine whether the sequence is arithmetic. If is arithmetic, find the common difference. $$ \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \dots $$

Determine whether the sequence is geometric. If is geometric, find the common ratio. $$ \frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \frac{1}{8}, \dots $$