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Math Help - Series and Exponential Growth

  1. #1
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    Series and Exponential Growth

    If you invest $1000 at 5.7% interest compounded quarterly, and you deposit $1000 each year, in how many years will you have a total of $1,000,000?

    It's not difficult to set up the problem without the additional $1000 each year, but with it, I'm lost.
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  2. #2
    Member mohammadfawaz's Avatar
    Joined
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    Lebanon - Beirut
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    Assuming you x_n dollars at the start of that year, let's try to find the amount we have at the end of the year:
    at the end of each quarter, we multiply by 1.057,
    hence, at the end of the year we have: x_{n+1} = x_{n}*1.057^4+1000 = 1.248x_n+1000. Our goal is to find n for which x_n\geq 1,000,000.
    This is difference equation which looks like: x_{n+1}-1.248x_{n}=1000 whose solution is of the form x_{n}=Ar^n+C (A, r and C are constants). To find these, solve first for x_{n+1}-1.248x_{n}=0. You get A and r. Then plug in what you got in the initial equation to find C.
    Now that you have the closed form, you can solve for x_{n}\geq 1,000,000
    hope this helps
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