Question

You have $23,474.12 in a brokerage account, and you plan to deposit an additional $5,000 at the end of every future year until your account totals $230,000. You expect to earn 13% annually on the account. How many years will it take to reach your goal? Round your answer to the nearest whole number.

years

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.

Future value of 23,474.12=23,474.12*(1.13)^n

Also:

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

=5000[(1.13)^n-1]/0.13

Hence

230,000=23,474.12*(1.13)^n+5000[(1.13)^n-1]/0.13

230,000=23,474.12*(1.13)^n+38461.5385[(1.13)^n-1]

230,000=23,474.12*(1.13)^n+38461.5385*(1.13)^n-38461.5385

(230,000+38461.5385)=(1.13)^n*[23,474.12+38461.5385]

(1.13)^n=(230,000+38461.5385)/(23,474.12+38461.5385)

Taking log on both sides;

n*log 1.13=log 4.33452304

n=log 4.33452304/log 1.13

=12 years(Approx).