Chapter 12

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

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

9–16 Determine whether the sequence is arithmetic. If it is arithmetic, find the common difference. $$5,8,11,14, \dots$$

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

\(1-12\) . Use Pascal's triangle to expand the expression. $$ \left(1+x^{3}\right)^{3} $$

Funding an Annuity 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 semi- annually. He is to draw semiannual payments until he reaches age \(65 .\) What is the amount of each payment?

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

9–16 Determine whether the sequence is arithmetic. If it is arithmetic, find the common difference. $$3,6,9,13, \dots$$

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

\(1-12\) . Use Pascal's triangle to expand the expression. $$ \left(\frac{1}{x}-\sqrt{x}\right)^{5} $$

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?

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

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

9–16 Determine whether the sequence is arithmetic. If it is arithmetic, find the common difference. $$2,4,8,16, \dots$$

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

\(1-12\) . Use Pascal's triangle to expand the expression. $$ \left(2+\frac{x}{2}\right)^{5} $$

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. $$ 27,-9,3,-1, \ldots $$

9–16 Determine whether the sequence is arithmetic. If it is arithmetic, find the common difference. $$2,4,6,8, \dots$$

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

\(13-20=\) Evaluate the expression. $$ \left(\begin{array}{l}{6} \\ {4}\end{array}\right) $$

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?

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. $$ \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \dots $$

9–16 Determine whether the sequence is arithmetic. If it is arithmetic, find the common difference. $$3, \frac{3}{2}, 0,-\frac{3}{2}, \dots$$

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

\(13-20=\) 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. $$ e^{2}, e^{4}, e^{6}, e^{8}, \dots $$

9–16 Determine whether the sequence is arithmetic. If it is arithmetic, find the common difference. $$\ln 2, \ln 4, \ln 8, \ln 16, \ldots$$

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

\(13-20=\) Evaluate the expression. $$ \left(\begin{array}{c}{100} \\ {98}\end{array}\right) $$

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?

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. $$ 1.0,1.1,1.21,1.331, \dots $$

9–16 Determine whether the sequence is arithmetic. If it is arithmetic, find the common difference. $$2.6,4.3,6.0,7.7, \ldots$$

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

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?

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

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

9–16 Determine whether the sequence is arithmetic. If it is arithmetic, find the common difference. $$\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \dots$$

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

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?

Show that \(8^{n}-3^{n}\) is divisible by 5 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 \(a_{n}=a r^{n-1} .\) $$ a_{n}=2(3)^{n} $$

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 \(a_{n}=a+(n-1) d .\) $$a_{n}=4+7 n$$

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. \(a_{n}=4 n+3\)

\(13-20=\) 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?

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