Question

You want $5000 4 years from nowfor a down payment for a car. How much money must be deposited monetly into an account earning 4.2% compounded monthly to achieve this goal?

Answer 1

$95.5090 must be deposited monthly

Explanation:

From the information given:

The annual interest rate (r) = 4.2% = 0.042

Let assume that an amount Y is deposited, then after one month, it will increase to:

`Y ( 1+ (0.042)/(12))`

The total amount after 4 years will be:

`= Y ( 1+ (0.042)/(12))^(48)+Y ( 1+ (0.042)/(12))^(47) +Y ( 1+ (0.042)/(12))^(46) +...+ Y ( 1+ (0.042)/(12))`

`= Y ( 1.0035)^(48)+Y ( 1.0035)^(47) +Y( 1.0035)^(46) +...+ Y( 1.0035)`

Using the sum of a geometric progression:

`= Y (1.0035) ( (1.0035^(48)-1))/((1.0035-1))`

`= Y (1.0035) ( (1.0035^(48)-1))/(0.0035)`

The above amount is then equal to $5000

i.e

`= Y (1.0035) ( (1.0035^(48)-1))/(0.0035) = 5000`

`Y = (5000* 0.0035)/((1.0035)(1.0035^(48)-1)) \\ \\ \mathbf{Y = \$95.5090}`