91Ó°ÊÓ

Use mathematical induction to prove that the sum of the cubes of any three consecutive natural numbers is a multiple of 9 .

For the sequence \(a_{1}, a_{2}, \ldots, a_{n}, \ldots,\) assume that \(a_{1}=1, a_{2}=1,\) and that for each \(n \in \mathbb{N}, a_{n+2}=\frac{1}{2}\left(a_{n+1}+\frac{2}{a_{n}}\right)\). (a) Calculate \(a_{3}\) through \(a_{6}\). (b) Prove that for each \(n \in \mathbb{N}, 1 \leq a_{n} \leq 2\).

(a) Prove that if \(n \in \mathbb{N},\) then there exists an odd natural number \(m\) and a nonnegative integer \(k\) such that \(n=2^{k} m\). (b) For each \(n \in \mathbb{N}\), prove that there is only one way to write \(n\) in the form described in Part (a). To do this, assume that \(n=2^{k} m\) and \(n=2^{q} p\) where \(m\) and \(p\) are odd natural numbers and \(k\) and \(q\) are nonnegative integers. Then prove that \(k=q\) and \(m=p\).

The Future Value of an Ordinary Annuity. For an ordinary annuity, \(R\) dollars is deposited in an account at the end of each compounding period. It is assumed that the interest rate, \(i,\) per compounding period for the account remains constant. Let \(S_{t}\) represent the amount in the account at the end of the \(t\) th compounding period. \(S_{t}\) is frequently called the future value of the ordinary annuity. So \(S_{1}=R\). To determine the amount after two months, we first note that the amount after one month will gain interest and grow to \((1+i) S_{1} .\) In addition, a new deposit of \(R\) dollars will be made at the end of the second month. So $$ S_{2}=R+(1+i) S_{1} $$ (a) For each \(n \in \mathbb{N},\) use a similar argument to determine a recurrence relation for \(S_{n+1}\) in terms of \(R, i,\) and \(S_{n}\). (b) By recognizing this as a recursion formula for a geometric series, use Proposition 4.16 to determine a formula for \(S_{n}\) in terms of \(R, i,\) and \(n\) that does not use a summation. Then show that this formula can be written as $$ S_{n}=R\left(\frac{(1+i)^{n}-1}{i}\right) $$ (c) What is the future value of an ordinary annuity in 20 years if \(\$ 200\) dollars is deposited in an account at the end of each month where the interest rate for the account is \(6 \%\) per year compounded monthly? What is the amount of interest that has accumulated in this account during the 20 years?