Question

You have $26,261.67 in a brokerage account, and you plan to deposit an additional $3,000 at the end of every future year until your account totals $250,000. You expect to earn 13% annually on the account. How many years will it take to reach your goal? Round your answer to two decimal places at the end of the calculations.

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 $26,261.67 is equal to

=26,261.67*(1.13)^n

Also:

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

=$3000[(1.13)^n-1]/0.13

Hence

250000 =26,261.67*(1.13)^n+$3000[(1.13)^n-1]/0.13

250,000=26,261.67*(1.13)^n+23076.92308[(1.13)^n-1]

250,000=26,261.67*(1.13)^n+23076.92308*(1.13)^n-23076.92308

(250,000+23076.92308)=(1.13)^n[26,261.67+23076.92308]

273076.92308=(1.13)^n*49338.59308

(1.13)^n=(273076.92308/49338.59308)

(1.13)^n=5.534752939

Taking log on both sides;

n*log (1.13)=log5.534752939

n=log5.534752939/log 1.13

=14 years(Approx).