Chapter 11

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

Find the first four terms and the 100 th term of the sequence. $$a_{n}=n^{n}$$

Determine whether the sequence is arithmetic. If it is arithmetic, find the common difference. $$3,6,9,13, \dots$$

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

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

Funding an Annuity A 55 -year-old man deposits \(\$ 50,000\) to fund an annuity with an insurance company. The money will be invested at \(8 \%\) per year, compounded semiannually. He is to draw semiannual payments until he reaches age \(65 .\) What is the amount of each payment?

Find the first four terms and the 100 th term of the sequence. $$a_{n}=3$$

Determine whether the sequence is arithmetic. If it is arithmetic, find the common difference. $$2,4,8,16, \dots$$

Use mathematical induction to prove that the formula is true for all natural numbers \(n\). $$\begin{aligned} 1 \cdot 2+2 \cdot 2^{2}+3 \cdot 2^{3}+4 \cdot 2^{4}+& \cdots+n \cdot 2^{n} \\ &=2\left[1+(n-1) 2^{n}\right] \end{aligned}$$

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

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

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

Financing a Car A woman wants to borrow \(\$ 12,000\) in order 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?

Determine whether the sequence is arithmetic. If it is arithmetic, find the common difference. $$2,4,6,8, \dots$$

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 it is geometric, find the common ratio. $$27,-9,3,-1, \ldots$$

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

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

Mortgage \(\quad\) What is the monthly payment on a 30 -year mortgage of \(\$ 80,000\) at \(9 \%\) interest? What is the monthly payment on this same mortgage if it is to be repaid over a 15-year period?

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

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

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

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

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?

Determine whether the sequence is arithmetic. If it is arithmetic, find the common difference. $$\ln 2, \ln 4, \ln 8, \ln 16, \dots$$

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

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

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

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

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?

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

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

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

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

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?

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

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

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

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

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

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?

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

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

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

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?

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. $$a_{n}=4 n+3$$

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

Evaluate the expression. $$\left(\begin{array}{l} 5 \\ 2 \end{array}\right)\left(\begin{array}{l} 5 \\ 3 \end{array}\right)$$

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?

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. $$a_{n}=n^{2}+n$$