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Math Help - Simple exponential function

  1. #1
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    Simple exponential function

    I have 4 values 200,500,900,1400 lets say there x and y increases by 1 each time. I need the formula for this so i can program it.

    I figured out that
    (100*(x+1))
    is the difference between the numbers just i cant get the rest...
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  2. #2
    Senior Member bkarpuz's Avatar
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    Quote Originally Posted by botanic View Post
    I have 4 values 200,500,900,1400 lets say there x and y increases by 1 each time. I need the formula for this so i can program it.

    I figured out that
    (100*(x+1))
    is the difference between the numbers just i cant get the rest...
    The values you gave come out froum the solution of the following difference equation:
    f(x)=f(x-1)+100*(x+1)\text{ for }n\in\mathbb{N}\text{ with the initial condition }f(0)=0.
    Then, we have
    f(x)-f(x-1)=100*(x+1).
    By summing up from i=1 to x, we get
    \sum\limits_{i=1}^{x}\big[f(i)-f(i-1)\big]=100\sum\limits_{i=1}^{x}(i+1).\qquad(*)
    Note that left-hand side of (*) is a telescopic series, hence we deduce that
    \Rightarrow f(x)-f(0)=100\sum\limits_{i=1}^{x}(i+1)
    ...................._ =100\Bigg(\sum\limits_{i=1}^{x}i+\sum\limits_{i=1}  ^{x}1\Bigg)
    ...................._ =100\Bigg(\frac{x(x+1)}{2}-x\Bigg),
    or simply, we have
    \Rightarrow f(x)=50x(x+3)
    for all x\in\mathbb{N} since f(0)=0.

    I hope this is what you want.
    Last edited by bkarpuz; September 14th 2008 at 11:21 PM. Reason: f is simplified, it was f(x)=100((x+1)(x+2)/2-1).
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