﻿ Loans and Investments (Part 2)

# Loans and Investments (Part 2)

## Saving for the Future

You are saving for the future. Your initial deposit is $\,\$4100\,.$Interest is being earned at an annual rate of$\,5\%\,,$compounded monthly. You will contribute an additional$\,\$120\,$ each month.

(a)
(b)  Write a recursive formula where $\,u_n\,$ gives the amount saved (principal plus interest) after $\,n\,$ months.
(c)  Then, find the amount saved (principal plus interest) after $\,7\,$ years.
(d)  Find the total amount of money you contributed (principal only) during these $\,7\,$ years.
(e)  Find the total interest earned during these $\,7\,$ years.

## Solution

### (a) Find the interest earned in the first month

The interest earned in the first month is: $$\cssId{sb18}{(\4{,}100)(\frac{0.05}{12}) = \17.08}$$

### (b) Write a recursive formula where $\,u_n\,$ gives the amount saved (principal plus interest) after $\,n\,$ months

The recursive formula is: \begin{align} &\cssId{sb21}{u_0 = 4100}\cr &\cssId{sb22}{u_n = (1 + \frac{0.05}{12})u_{n-1} + 120\,,\ \ \text{ for } n \ge 1} \end{align}

### (c) Find the amount saved (principal plus interest) after $\,7\,$ years

Note that $\,7\,$ years is $\,7(12) = 84\,$ months.

From the calculator, or from the form below: $$\cssId{sb26}{u_{84} = 17853.39}$$

Thus, you have saved (principal plus interest) $\,\$17{,}853.39\,$after$\,7\,$years. The form below computes the amount saved (principal plus interest) after$\,n\,$equal monthly payments: • Your initial deposit is$\,u_0\,$(in dollars). That is,$\,u_0\,$is the amount saved at time zero (the start of your savings program). • Interest is being earned at an annual rate of$\,i\%\,,$compounded monthly. For example, if the interest rate is$\,5\%\,,$then$\,i = 5\,.$That is,$\,i\,$does not include the percent sign. • You are contributing an additional$\,\$C\,$ each month.