Question

You have $33,749.85 in a brokerage account, and you plan to deposit an additional $4,000 at the end of every future year until your account totals $250,000. You expect to earn 10.2% annually on the account. How many years will it take to reach your goal? Round your answer to the nearest whole number.

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 33,749.85 =33,749.85 *(1.102)^n

Also:

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

=$4000[(1.102)^n-1]/0.102

Hence total future value=33,749.85 *(1.102)^n+$4000[(1.102)^n-1]/0.102

250,000=33,749.85 *(1.102)^n+$4000[(1.102)^n-1]/0.102

250,000=33,749.85 *(1.102)^n+$39215.68627[(1.102)^n-1]

250,000=33,749.85 *(1.102)^n+$39215.68627*(1.102)^n-39215.68627

(250,000+39215.68627)=(1.102)^n[33,749.85+39215.68627]

(250,000+39215.68627)/[33,749.85+39215.68627]=(1.102)^n

(1.102)^n=3.963730016

Taking log on both sides;

n*log 1.102=log 3.963730016

n=log 3.963730016/log 1.102

which is equal to

=14 years(Approx).