Question

You deposited $500 each month into your new ira account at the beginning of each month. there was no previous balance. you were fortunate to have earned 5.2% (apr) the entire time, and your account balance grew to $247,014! for how many years did you make the payments?

Answer 1

This question would be solved using a BA II plus calculator as no educational institution would require a manual evaluation, keystrokes on calculator are as follows: [2ND] PMT [2ND] ENTER ***These 4 keystrokes set the device to beginning of period compounding instead of end of period. We do this because the question states that the $500 deposits are made at the beginning of the month and therefore are entitled to the entire months worth of interest compounding. If done correctly, you will see the letters BGN across the top right of the display. Keystrokes continue as follows: PV 0 PMT 500 FV -247014 VERY IMPORTANT TO ENTER FV AS NEGATIVE NUMBER I/Y 5.2 / 12 Because its a monthly payment, the annual apr given is divided by 12 CPT N DISPLAY WILL READ 264. This means 264 monthly payments were made. To express this in years we simply divide this number by 12 to get: 264/12 = 22. We report that we made payments for 22 years.

Answer 2

Let n = the number of years that payments were made.

Assume that the compounding interval is 12 (monthly).
The 1st payment acquires the value

`500(1+ (0.052)/(12) )^(12n) = 500(1.00433)^(12n)`
The 2nd payment acquires the value

`500(1.00433)^(12n-1)`
and so on.

The value of the payments forms a geometric series

`500(1.00433),\, 500(1.00433)^(2),\, 500(1.00433)^(3) , \, ... \, 500(1.00433)^(12n)`

The sum of the geometric series is 247014, therefore

`(500*1.00433(1 - 1.00433^(12n)))/(1-1.00433)=247014 \\\\ 1-1.00433^(12n) = -2.1316 \\\\ 1.00433^(12n) = 3.1316 \\\\ 12n = (ln(3.1316))/(ln(1.00433)) =264 \\\\ n = 22`

Answer: 22 years