Question

You have $10,000 in a brokerage account, and you plan to deposit an additional $7,500 at the end of every future year until your account totals $500,000. You expect to earn 9% annually on the account. How many years will it take to reach your goal?

Round you answer UP to the nearest whole number of years. (Example 5.1 = 6)

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 10,000=10,000*(1.09)^n

Also:

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

=7500[(1.09)^n-1]/0.09

Hence

500,000=10,000*(1.09)^n+7500[(1.09)^n-1]/0.09

500,000=10,000*(1.09)^n+83333.3333*[(1.09)^n-1]

500,000=10,000*(1.09)^n+83333.3333*(1.09)^n-83333.3333

(500,000+83333.3333)=(1.09)^n[10,000+83333.3333]

(500,000+83333.3333)/(10,000+83333.3333)=(1.09)^n

(1.09)^n=6.25

Taking log on both sides;

n*log 1.09=log 6.25

n=log 6.25/log 1.09

=22 years(Approx)