Chapter 12, Problem 7P

(a)

To determine

Toexplain: The recursive formula gives the final amount in Vitoria’s CD.

Explanation of Solution

Given:The recursive formula gives the amount $V_n$ in Victoria’s CD.

  $V_n=1.05V_{n-1}+500n$

The recursive formula, $V_n=1.05V_{n-1}+500n$

Where, $n$ represents the number of years.

  $V_{n-1}$ represents the previous year amount in Victoria’s CD.

Interest of one-year CD with 5%

So, next year amount = $1.05V_{n-1}$

Victoria also decides to add $500 each year extra.

So, $500n$ represents the nth year deposit in CD account.

Thus, the recursive formula represent the amount in Victoria’s CD account.

  $V_n$ is final amount after n years.

(b)

To determine

To compare: The amount of Victoria’s and Ursula

Explanation of Solution

Given: The recursive formula gives the amount $V_n$ in Victoria’s CD.

  $V_n=1.05V_{n-1}+500n$

Recursive formula for Ursula, $U_n=1.05U_{n-1}+0.1U_{n-1}$

First make table for some years amount of Vitoria and Ursula.

  $\begin{array}{ccc}n& U_n& V_n\\ 0& 5000& 5000\\ 1& 5750& 5750\\ 2& 6612.5& 7037.5\\ 3& 7604.4& 8889.4\\ 4& 8745& 11334\\ 5& 10057& 14401\\ 6& 11565& 18121\end{array}$

First year both are same amount.

In second year Victoria amount is more than Ursula. It increases continuously every year.

  

After 13 years Ursula amount is equal to Victoria amount in 6^th year.

Hence , Ursula is growing but very slow as compare to Victoria.