Question

You have $34,717.34 in a brokerage account, and you plan to deposit an additional $3,000 at the end of every future year until your account totals $260,000. 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.

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 34,717.34=34,717.34*(1.13)^n

Also:

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

=3000[(1.13)^n-1]/0.13

Hence

260,000= 34,717.34*(1.13)^n+3000[(1.13)^n-1]/0.13

260,000= 34,717.34*(1.13)^n+23076.9231*[(1.13)^n-1]

260,000= 34,717.34*(1.13)^n+23076.9231*(1.13)^n-23076.9231

(260,000+23076.9231)=(1.13)^n[34,717.34+23076.9231]

(260,000+23076.9231)/(34,717.34+23076.9231)=(1.13)^n

(1.13)^n=4.89801077

Taking log on both sides;

n*log 1.13=log 4.89801077

n=log 4.89801077/log 1.13

=13 years(Approx)