Question

You have $5,547.44 in a brokerage account, and you plan to deposit an additional $6,000 at the end of every future year until your account totals $240,000. 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.

Hence future value of 5,547.44=5,547.44*(1.13)^n

Also:

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

=6000[(1.13)^n-1]/0.13

Hence total future value=5,547.44*(1.13)^n+6000[(1.13)^n-1]/0.13

240,000=5,547.44*(1.13)^n+6000[(1.13)^n-1]/0.13

240,000=5,547.44*(1.13)^n+46153.84615[(1.13)^n-1]

240,000=5,547.44*(1.13)^n+46153.84615*(1.13)^n-46153.84615

(240,000+46153.84615)=(1.13)^n[5547.44+46153.84615]

(1.13)^n=(240,000+46153.84615)/(5547.44+46153.84615)

Taking log on both sides;

n*log 1.13=log 5.534752952

n=log 5.534752952/log 1.13

=14 years(Approx).