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Math Help - Compound Interest

  1. #1
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    Compound Interest

    How do I convert P(1 + r)3 and P( 1 + r)4, etc. to the expanded form?
    I get confused by the factored version P(1 + r)n. This does not help me to understand.

    For example, P(1 + r)2 equals (P + rP) + r(P + rP). Show how this occurs with three and four years.
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  2. #2
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    Quote Originally Posted by jhonydeep3305 View Post
    How do I convert P(1 + r)3 and P( 1 + r)4, etc. to the expanded form?
    I get confused by the factored version P(1 + r)n. This does not help me to understand.

    For example, P(1 + r)2 equals (P + rP) + r(P + rP). Show how this occurs with three and four years.
    I don't really understand what you mean...

    But if it helps...

    Compound interest can be written as a series of recursive simple interest functions with interest being charged on interest in each time period. This can then be converted to a closed form, which you know as the compound interest formula.


    You should know the simple interest formula

    I = PrT

    so the amount accumulated is

    A = P + I.


    If we assume that we recalculate the simple interest every time period, then T = 1 and we can rewrite A as a new principal for the next calculation.


    So...

    P_1 = P_0 + I_0

     = P_0 + P_0r

     = P_0(1 + r).


    P_2 = P_1 + I_1

     = P_1 + P_1r

     = P_1(1 + r)

     = P_0(1 + r)(1 + r)

     = P_0(1 + r)^2.


    P_3 = P_2 + I_2

     = P_2 + P_2r

     = P_2(1 + r)

     = P_0(1 + r)^2(1 + r)

     = P_0(1 + r)^3.


    P_4 = P_3 + I_3

     = P_3 + P_3r

     = P_3(1 + r)

     = P_0(1 + r)^3(1 + r)

     = P_0(1 + r)^4.



    I think you can now see that after n time periods, the amount accumulated equates to

    A = P(1 + r)^n, with I = A - P.
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  3. #3
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    Quote Originally Posted by jhonydeep3305 View Post
    How do I convert P(1 + r)3 and P( 1 + r)4, etc. to the expanded form?
    I get confused by the factored version P(1 + r)n. This does not help me to understand.

    For example, P(1 + r)2 equals (P + rP) + r(P + rP). Show how this occurs with three and four years.
    To start, P(1 + r)^n means what $P deposited today will be worth in n years at annual rate r%.

    EXAMPLE: if $1000 is deposited today, rate is 9%, what will it accumulate to over 3 years?
    1000(1 + .09)^3 = 1295.03

    The calculation is 1000 * 1.09 * 1.09 * 1.09 which results in 1295.03

    > For example, P(1 + r)^2 equals (P + rP) + r(P + rP). Show how this occurs with three and four years.
    I also don't understand what you're getting at here; seems irrelevant....
    (1 + r)^2 = (1 + r) * (1 + r) = r^2 + 2r + 1; so P(r^2 + 2r + 1)
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