Question

TIME TO REACH A FINANCIAL GOAL You have $65,767.89 in a brokerage account, and you plan to deposit an additional $5,000 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 UP to the nearest year. (Example 5.01 years = 6 years) Your answer should include numerical value only.

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 A for 65,767.89 =65,767.89 *(1.11)^n

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

=$5000[(1.11)^n-1]/0.11

Hence

250,000=65,767.89 *(1.11)^n+$5000[(1.11)^n-1]/0.11

250,000=65,767.89 *(1.11)^n+$45,454.54545[(1.11)^n-1]

250,000=65,767.89 *(1.11)^n+$45,454.54545(1.11)^n-45,454.54545

(250,000+$45,454.54545)=(1.11)^n[65,767.89+45,454.54545]

(250,000+$45,454.54545)/[65,767.89+45,454.54545]=(1.11)^n

(1.11)^n=2.656429382

Taking log on both sides;

n*log 1.11=log2.656429382

n=log2.656429382/log 1.11

9 years(Approx).