Chapter 13

\(19-24\) . 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. $$ a_{n}=n^{2}+n $$

Evaluate the expression. $$ \left(\begin{array}{c}{10} \\ {5}\end{array}\right) $$

Show that \(3^{2 n}-1\) is divisible by 8 for all natural numbers \(n\)

Mortgage A couple can afford to make a monthly mortgage payment of \(\$ 650\) . If the mortgage rate is 9\(\%\) and the couple intends to secure a 30-year mortgage, how much can they borrow?

21-26 \(\approx\) Find the first five terms of the sequence, and determine whether it is arithmetic. If it is arithmetic, find the common difference, and express the \(n\) th term of the sequence in the standard form \(a_{n}=a+(n-1) d\) $$ a_{n}=4+7 n $$

Find the first five terms of the sequence, and determine whether it is geometric. If is geometric, find the common ratio, and express the \(n\) th term of the sequence in the standard form \(a_{n}=a r^{n-1} .\) $$ a_{n}=2(3)^{n} $$

\(19-24\) . 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. $$ a_{n}=\frac{12}{n} $$

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

Financing a Car Jane agrees to buy a car for a down payment of \(2000 and payments of \)220 per month for 3 years. If the interest rate is 8% per year, compounded monthly, what is the actual purchase price of her car?

21-26 \(\approx\) Find the first five terms of the sequence, and determine whether it is arithmetic. If it is arithmetic, find the common difference, and express the \(n\) th term of the sequence in the standard form \(a_{n}=a+(n-1) d\) $$ a_{n}=4+2^{n} $$

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

\(19-24\) . 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. $$ a_{n}=4-2(-1)^{n} $$

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

Financing a Ring Mike buys a ring for his fiancee by paying $30 a month for one year. If the interest rate is 10% per year, compounded monthly, what is the price of the ring?

21-26 \(\approx\) Find the first five terms of the sequence, and determine whether it is arithmetic. If it is arithmetic, find the common difference, and express the \(n\) th term of the sequence in the standard form \(a_{n}=a+(n-1) d\) $$ a_{n}=\frac{1}{1+2 n} $$

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

\(19-24\) . 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. $$ a_{n}=\frac{1}{a_{n-1}} \quad \text { and } \quad a_{1}=2 $$

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

Mortgage A couple secures a 30 -year loan of \(\$ 100,000\) at 9\(\frac{3}{4} \%\) per year, compounded monthly, to buy a house. (a) What is the amount of their monthly payment? (b) What total amount will they pay over the 30 -year period? (c) If, instead of taking the loan, the couple deposits the monthly payments in an account that pays 9\(\frac{3}{4} \%\) interest per year, compounded monthly, how much will be in the account at the end of the 30 -year period?

21-26 \(\approx\) Find the first five terms of the sequence, and determine whether it is arithmetic. If it is arithmetic, find the common difference, and express the \(n\) th term of the sequence in the standard form \(a_{n}=a+(n-1) d\) $$ a_{n}=1+\frac{n}{2} $$

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

\(19-24\) . 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. $$ a_{n}=a_{n-1}-a_{n-2} \quad \text { and } \quad a_{1}=1, a_{2}=3 $$

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

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

Mortgage A couple needs a mortgage of \(\$ 300,000\) . Their mortgage broker presents them with two options: a 30 -year mortgage at 6\(\frac{1}{2} \%\) interest or a 15 -year mortgage at 5\(\frac{3}{4} \%\) interest. (a) Find the monthly payment on the 30 -year mortgage and on the 15 -year mortgage. Which mortgage has the larger monthly payment? (b) Find the total amount to be paid over the life of each loan. Which mortgage has the lower total payment over its lifetime?

21-26 \(\approx\) Find the first five terms of the sequence, and determine whether it is arithmetic. If it is arithmetic, find the common difference, and express the \(n\) th term of the sequence in the standard form \(a_{n}=a+(n-1) d\) $$ a_{n}=6 n-10 $$

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

\(25-32\) . Find the \(n\) th term of a sequence whose first several terms are given. $$ 2,4,8,16, \dots $$

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

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

21-26 \(\approx\) Find the first five terms of the sequence, and determine whether it is arithmetic. If it is arithmetic, find the common difference, and express the \(n\) th term of the sequence in the standard form \(a_{n}=a+(n-1) d\) $$ a_{n}=3+(-1)^{n} n $$

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

\(25-32\) . Find the \(n\) th term of a sequence whose first several terms are given. $$ -\frac{1}{3}, \frac{1}{9},-\frac{1}{27}, \frac{1}{81}, \dots $$

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

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

Interest Rate Janet’s payments on her \(12,500 car are \)420 a month for 3 years. Assuming that interest is compounded monthly, what interest rate is she paying on the car loan?

\(27-36\) . Determine the common difference, the fifth term, the nth term, and the 100 th term of the arithmetic sequence. $$ 2,5,8,11, \dots $$

Determine the common ratio, the fifth term, and the nth term of the geometric sequence. $$ 2,6,18,54, \dots $$

\(25-32\) . Find the \(n\) th term of a sequence whose first several terms are given. $$ 1,4,7,10, \ldots $$

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

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

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.

\(27-36\) . Determine the common difference, the fifth term, the nth term, and the 100 th term of the arithmetic sequence. $$ 1,5,9,13, \dots $$

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

\(25-32\) . Find the \(n\) th term of a sequence whose first several terms are given. $$ 5,-25,125,-625, \dots $$

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

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

\(27-36\) . Determine the common difference, the fifth term, the nth term, and the 100 th term of the arithmetic sequence. $$ 4,9,14,19, \dots $$

Determine the common ratio, the fifth term, and the nth term of the geometric sequence. $$ 0.3,-0.09,0.027,-0.0081, \dots $$