Question

You have $23,199.50 in a brokerage account, and you plan to deposit an additional $4,000 at the end of every future year until your account totals $200,000. You expect to earn 10% 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. years

Answer 1

Let 'n' be the time period

We use the formula:
A=P(1+r/100)^n
where
A=future value
P=present value
r=rate of interest
n=time period

A=$23199.5(1.1)^n

Also;

Future value of annuity of $4000=Annuity[(1+rate)^time period-1]/rate

=$4000[(1.1)^n-1]/0.1

Hence

200,000=$23199.5(1.1)^n+$4000[(1.1)^n-1]/0.1

200,000=23199.5(1.1)^n+40000[(1.1)^n-1]

200,000=23199.5(1.1)^n+40000(1.1)^n-40000

(200,000+40000)=23199.5(1.1)^n+40000(1.1)^n

240000=(1.1)^n[23199.5+40000]

240,000=63199.5(1.1)^n

(240000/63199.5)=(1.1)^n

Taking log on both sides;

log(240000/63199.5)=n*log 1.1

n=log(240000/63199.5)/log 1.1

which is equal to

=14 years.