You have 2626167 in a brokerage account and you plan to depo

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.

Solution

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).