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12 March, 04:30

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

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  1. 12 March, 07:12
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    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).
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