You have 4412895 in a brokerage account and you plan to depo

You have $44,128.95 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 11% 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.

___ years

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 $44128.95=$44128.95*(1.11)^n

Also:

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

=$3000[(1.11)^n-1]/0.11

Hence

250000=$44128.95*(1.11)^n+$3000[(1.11)^n-1]/0.11

250,000=$44128.95*(1.11)^n+27272.72727[(1.11)^n-1]

250,000=$44128.95*(1.11)^n+27272.72727*(1.11)^n-27272.72727

(250,000+27272.72727)=(1.11)^n[44128.95+27272.72727]

277,272.7273=(1.11)^n*71401.67727

(1.11)^n=(277,272.7273/71401.67727)

Taking log on both sides;

n*log 1.11=log 3.883280308

n=log 3.883280308/log 1.11

=13 years(Approx).