You have $22,566.87 in a brokerage account, and you plan to deposit an additional $5,000 at the end of every future year until your account totals $280,000. You expect to earn 11% annually on the account. How many years will it take to reach your goal

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 $22,566.87=$22,566.87 * (1.11) ^n

Also:

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

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

Hence

280,000=22,566.87 * (1.11) ^n+$5000[ (1.11) ^n-1]/0.11

280,000=22,566.87 * (1.11) ^n+$45,454.55[ (1.11) ^n-1]

280,000=22,566.87 * (1.11) ^n+$45,454.55 * (1.11) ^n-45,454.55

(280,000+45,454.55) = (1.11) ^n[22566.87+45,454.55]

(1.11) ^n = (280,000+45,454.55) / [22566.87+45,454.55]

(1.11) ^n=4.784589431

Taking log on both sides;

n*log 1.11=log 4.784589431

n=log 4.784589431/log 1.11

=15 years (Approx).