Question

You have $42,912.42 in a brokerage account, and you plan to deposit an additional $7,500 at the end of every future year until your account totals $375,000. You expect to earn 7.5% annually on the account. How many years will it take to reach your goal? Round your answer to the nearest whole number.

Answer 1

We use the formula:
A=P(1+r/100)^n
where
A=future value
P=present value
r=rate of interest
n=time period.

Hence future value of $42,912.42 =$42,912.42 *(1.075)^n

Future value of annuity=Annuity[(1+rate)^time period-1]/rate

=$7500[(1.075)^n-1]/0.075

Hence

375000=$42,912.42 *(1.075)^n+$7500[(1.075)^n-1]/0.075

375000=$42,912.42 *(1.075)^n+$100,000[(1.075)^n-1]

375000=$42,912.42 *(1.075)^n+$100,000*(1.075)^n-100,000

(375000+100,000)=(1.075)^n[42,912.42+100,000]

(375000+100,000)/[42,912.42+100,000]=(1.075)^n

(1.075)^n=3.323713922

Taking log on both sides;

n*log 1.075=log 3.323713922

n=log 3.323713922/log 1.075

=17 years(Approx).