Question

Number of Periods of an Annuity You have $57,170.68 in a brokerage account, and you plan to deposit an additional $3,000 at the end of every future year until your account totals $425,000. You expect to earn 9.9% 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.

Future value of 57,170.68=57,170.68*(1.099)^n

Also:

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

=3000[(1.099)^n-1]/0.099

425,000=57,170.68*(1.099)^n+3000[(1.099)^n-1]/0.099

425,000=57,170.68*(1.099)^n+30303.0303[(1.099)^n-1]

425,000=57,170.68*(1.099)^n+30303.0303*(1.099)^n-30303.0303

(425000+30303.0303)=(1.099)^n(30303.0303+57,170.68)

(425000+30303.0303)/(30303.0303+57,170.68)=(1.099)^n

(1.099)^n=5.20502707

Taking log on both sides;

n*log 1.099=log 5.20502707

n=log 5.20502707/log 1.099

=17 years(Approx)