Question

You have $32,506.07 in a brokerage account, and you plan to deposit an additional $6,000 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 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 32,506.07=32,506.07*(1.11)^n

Also:

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

=6000[(1.11)^n-1]/0.11

Hence

250,000=32,506.07*(1.11)^n+6000[(1.11)^n-1]/0.11

250,000=32,506.07*(1.11)^n+54545.4545*[(1.11)^n-1]

250,000=32,506.07*(1.11)^n+54545.4545*(1.11)^n-54545.4545

(250,000+54545.4545)=(1.11)^n[54545.4545+32,506.07]

(250,000+54545.4545)/[54545.4545+32,506.07]=(1.11)^n

(1.11)^n=3.49845056

Taking log on both sides;

n*log 1.11=log 3.49845056

n=log 3.49845056/log 1.11

=12 years(Approx)