You have $49,279.24 in a brokerage account, and you plan to deposit an additional $3,000 at the end of every future year until your account totals $270,000. You expect to earn 11% annually on the account. How many years will it take to reach your goal? Round your answer to the nearest whole number.
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 49,279.24 =49,279.24 *(1.11)^n
Also:
Future value of annuity=Annuity[(1+rate)^time period-1]/rate
=3000[(1.11)^n-1]/0.11
Hence
270,000=49,279.24 *(1.11)^n+3000[(1.11)^n-1]/0.11
270,000=49,279.24 *(1.11)^n+27272.7273*[(1.11)^n-1]
270,000=49,279.24 *(1.11)^n+27272.7273*(1.11)^n-27272.7273
(270,000+27272.7273)=(1.11)^n[27272.7273+49,279.24]
(1.11)^n=(270,000+27272.7273)/[27272.7273+49,279.24]
Taking log on both sides;
n*log 1.11=log 3.88328005
n=log 3.88328005/log 1.11
=13 years.
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