Chapter 12

Find the first five terms of the sequence and determine if it is geometric. If it is geometric, find the common ratio and express the \(n\)th term of the sequence in the standard form \(a_{n}=a r^{n-1} .\) $$ a_{n}=4+3^{n} $$

17-22 Find the first five terms of the sequence and determine if it is arithmetic. If is arithmetic, find the common difference and express the \(n\)th term of the sequence in the standard form \(a_{n}=a+(n-1) d .\) $$a_{n}=4+2^{n}$$

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. \(a_{n}=n^{2}+n\)

\(13-20=\) Evaluate the expression. $$ \left(\begin{array}{l}{5} \\ {0}\end{array}\right)+\left(\begin{array}{l}{5} \\\ {1}\end{array}\right)+\left(\begin{array}{l}{5} \\\ {2}\end{array}\right)+\left(\begin{array}{l}{5} \\\ {3}\end{array}\right)+\left(\begin{array}{l}{5} \\\ {4}\end{array}\right)+\left(\begin{array}{l}{5} \\ {5}\end{array}\right) $$

Prove that \(n<2^{n}\) for all natural numbers \(n.\)

Find the first five terms of the sequence and determine if it is geometric. If it is geometric, find the common ratio and express the \(n\)th term of the sequence in the standard form \(a_{n}=a r^{n-1} .\) $$ a_{n}=\frac{1}{4^{n}} $$

17-22 Find the first five terms of the sequence and determine if it is arithmetic. If is arithmetic, find the common difference and express the \(n\)th term of the sequence in the standard form \(a_{n}=a+(n-1) d .\) $$a_{n}=\frac{1}{1+2 n}$$

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. \(a_{n}=\frac{12}{n}\)

\(13-20=\) Evaluate the expression. $$ \left(\begin{array}{l}{5} \\ {0}\end{array}\right)-\left(\begin{array}{l}{5} \\\ {1}\end{array}\right)+\left(\begin{array}{l}{5} \\\ {2}\end{array}\right)-\left(\begin{array}{l}{5} \\\ {3}\end{array}\right)+\left(\begin{array}{l}{5} \\\ {4}\end{array}\right)-\left(\begin{array}{l}{5} \\ {5}\end{array}\right) $$

Prove that \((n+1)^{2}<2 n^{2}\) for all natural numbers \(n \geq 3.\)

Find the first five terms of the sequence and determine if it is geometric. If it is geometric, find the common ratio and express the \(n\)th term of the sequence in the standard form \(a_{n}=a r^{n-1} .\) $$ a_{n}=(-1)^{n} 2^{n} $$

17-22 Find the first five terms of the sequence and determine if it is arithmetic. If is arithmetic, find the common difference and express the \(n\)th term of the sequence in the standard form \(a_{n}=a+(n-1) d .\) $$a_{n}=1+\frac{n}{2}$$

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. \(a_{n}=4-2(-1)^{n}\)

\(21-24\) Use the Binomial Theorem to expand the expression. $$ (x+2 y)^{4} $$

Interest Rate An item at a department store is priced at \(\$ 189.99\) and can be bought by making 20 payments of \(\$ 10.50 .\) Find the interest rate, assuming that interest is compounded monthly.

Prove that if \(x>-1,\) then \((1+x)^{n} \geq 1+n x\) for all natural numbers \(n .\)

Find the first five terms of the sequence and determine if it is geometric. If it is geometric, find the common ratio and express the \(n\)th term of the sequence in the standard form \(a_{n}=a r^{n-1} .\) $$ a_{n}=\ln \left(5^{n-1}\right) $$

17-22 Find the first five terms of the sequence and determine if it is arithmetic. If is arithmetic, find the common difference and express the \(n\)th term of the sequence in the standard form \(a_{n}=a+(n-1) d .\) $$a_{n}=6 n-10$$

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. \(a_{n}=\frac{1}{a_{n-1}} \quad\) and \(\quad a_{1}=2\)

\(21-24\) Use the Binomial Theorem to expand the expression. $$ (1-x)^{5} $$

Show that \(100 n \leq n^{2}\) for all \(n \geq 100.\)

Find the first five terms of the sequence and determine if it is geometric. If it is geometric, find the common ratio and express the \(n\)th term of the sequence in the standard form \(a_{n}=a r^{n-1} .\) $$ a_{n}=n^{n} $$

17-22 Find the first five terms of the sequence and determine if it is arithmetic. If is arithmetic, find the common difference and express the \(n\)th term of the sequence in the standard form \(a_{n}=a+(n-1) d .\) $$a_{n}=3+(-1)^{n} n$$

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. \(a_{n}=a_{n-1}-a_{n-2} \quad\) and \(\quad a_{1}=1, a_{2}=3\)

\(21-24\) Use the Binomial Theorem to expand the expression. $$ \left(1+\frac{1}{x}\right)^{6} $$

An Annuity That Lasts Forever An annuity in perpetuity is one that continues forever. Such annuities are useful in setting up scholarship funds to ensure that the award continues. (a) Draw a time line (as in Example 1 ) to show that to set up an annuity in perpetuity of amount \(R\) per time period, the amount that must be invested now is $$ A_{p}=\frac{R}{1+i}+\frac{R}{(1+i)^{2}}+\frac{R}{(1+i)^{3}}+\cdots+\frac{R}{(1+i)^{n}}+\cdots $$ where \(i\) is the interest rate per time period. (b) Find the sum of the infinite series in part (a) to show that $$ A_{p}=\frac{R}{i} $$ (c) How much money must be invested now at 10\(\%\) per year, compounded annually, to provide an annuity in perpetuity of \(\$ 5000\) per year? The first payment is due in one year. (d) How much money must be invested now at 8\(\%\) per year, compounded quarterly, to provide an annuity in perpetuity of \(\$ 3000\) per year? The first payment is due in one year.

Determine the common ratio, the fifth term, and the \(n\)th term of the geometric sequence. $$ 2,6,18,54, \dots $$

Let \(a_{n+1}=3 a_{n}\) and \(a_{1}=5 .\) Show that \(a_{n}=5 \cdot 3^{n-1}\) for all natural numbers \(n .\)

23-32 me the common difference, the fifth term, the \(n\)th term, and the 100th term of the arithmetic sequence. $$2,5,8,11, \dots$$

Find the \(n\)th term of a sequence whose first several terms are given. \(2,4,8,16, \dots\)

\(21-24\) Use the Binomial Theorem to expand the expression. $$ \left(2 A+B^{2}\right)^{4} $$

Amortizing a Mortgage When they bought their house, John and Mary took out a \(\$ 90,000\) mortgage at 9\(\%\) interest, repayable monthly over 30 years. Their payment is \(\$ 724.17\) per month (check this using the formula in the text). The bank gave them an amortization schedule, which is a table showing how much of each payment is interest, how much goes toward the principal, and the remaining principal after each payment. The table below shows the first few entries in the amortization schedule. $$ \begin{array}{|c|c|c|c|c|}\hline \text { Payment } & {\text { Total }} & {\text { Interest }} & {\text { Principal }} & {\text { Remaining }} \\\ {\text { number }} & {\text { payment }} & {\text { payment }} & {\text { payment }} & {\text { principal }} \\ \hline 1 & {724.17} & {675.00} & {49.54} & {89,950.83} \\ {2} & {724.17} & {674.63} & {49.54} & {89,901.29} \\\ {3} & {724.17} & {674.26} & {49.91} & {89,851.38} \\ {4} & {724.17} & {673.89} & {50.28} & {89,801.10} \\ \hline\end{array} $$ After 10 years they have made 120 payments and are wondering how much they still owe, but they have lost the amortization schedule. (a) How much do John and Mary still owe on their mortgage? [Hint: The remaining balance is the present value of the 240 remaining payments. (b) How much of their next payment is interest and how much goes toward the principal? [Hint: Since 9\(\% \div\) \(12=0.75 \%\) , they must pay 0.75\(\%\) of the remaining principal in interest each month.

Determine the common ratio, the fifth term, and the \(n\)th term of the geometric sequence. $$ 7, \frac{14}{3}, \frac{28}{9}, \frac{56}{27}, \dots $$

A sequence is defined recursively by \(a_{n+1}=3 a_{n}-8\) and \(a_{1}=4 .\) Find an explicit formula for \(a_{n}\) and then use mathematical induction to prove that the formula you found is true.

23-32 me the common difference, the fifth term, the \(n\)th term, and the 100th term of the arithmetic sequence. $$1,5,9,13, \dots$$

Find the \(n\)th term of a sequence whose first several terms are given. \(-\frac{1}{3}, \frac{1}{9},-\frac{1}{27}, \frac{1}{81}, \dots\)

Find the first three terms in the expansion of \((x+2 y)^{20}\)

Determine the common ratio, the fifth term, and the \(n\)th term of the geometric sequence. $$ 0.3,-0.09,0.027,-0.0081, \ldots $$

Show that \(x-y\) is a factor of \(x^{n}-y^{n}\) for all natural numbers \(n .\) \(\left[\text { Hint: } x^{k+1}-y^{k+1}=x^{k}(x-y)+\left(x^{k}-y^{k}\right) y\right]\)

23-32 me the common difference, the fifth term, the \(n\)th term, and the 100th term of the arithmetic sequence. $$4,9,14,19, \dots$$

Find the \(n\)th term of a sequence whose first several terms are given. \(1,4,7,10, \dots\)

Find the first four terms in the expansion of \(\left(x^{1 / 2}+1\right)^{30}\)

Determine the common ratio, the fifth term, and the \(n\)th term of the geometric sequence. $$ 1, \sqrt{2}, 2,2 \sqrt{2}, \ldots $$

Show that \(x+y\) is a factor of \(x^{2 n-1}+y^{2 n-1}\) for all natural numbers \(n .\)

23-32 me the common difference, the fifth term, the \(n\)th term, and the 100th term of the arithmetic sequence. $$11,8,5,2, \dots$$

Find the \(n\)th term of a sequence whose first several terms are given. \(5,-25,125,-625, \dots\)

Find the last two terms in the expansion of \(\left(a^{2 / 3}+a^{1 / 3}\right)^{25}\)

Determine the common ratio, the fifth term, and the \(n\)th term of the geometric sequence. $$ 144,-12,1,-\frac{1}{12}, \dots $$

23-32 me the common difference, the fifth term, the \(n\)th term, and the 100th term of the arithmetic sequence. $$-12,-8,-4,0, \dots$$

Find the \(n\)th term of a sequence whose first several terms are given. \(1, \frac{3}{4}, \frac{5}{9}, \frac{7}{16}, \frac{9}{25}, \dots\)