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Math Help - Proof question.

  1. #1
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    Proof question.

    How do you prove this?

    Many thanks!
    Attached Thumbnails Attached Thumbnails Proof question.-provethisplease.jpg  
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  2. #2
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    Hi

    (1-x)\sum\limits_{k=0}^{N}x^k=\sum\limits_{k=0}^{N}x^  k-\sum\limits_{k=1}^{N+1}x^{k}=1-x^{N+1}

    If x\neq 1 then x-1\neq 0, and you can divide by 1-x.
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  3. #3
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    Quote Originally Posted by bluebiro View Post
    How do you prove this?

    Many thanks!
    Let S = 1 + x + x^2 + ... + x^N

    Then xS = x + x^2 + ....+ x^N + x^{N+1}

    Now subtract the two equations:

    S - xS = (1 + x + x^2+ ....+ x^N) - (x + x^2 + ....+ x^N + x^{N+1})

    Group terms now, so that they cancel,

    (1 - x)S = 1 +(x - x) + (x^2 - x^2)+ ....+(x^N -  x^N) - x^{N+1}

    After canceling, we have:

    (1 - x)S = 1 - x^{N+1}

    Now since x \neq 1, divide by (1 - x) on both sides and we are done!

    NOTE: This is exactly what clic-clac did, but I have explicitly written down the terms so that you can "see" the solution clearly. Try reading through clic-clac's post, to learn elegant ways of writing proofs involving sums.
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  4. #4
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    \sum\limits_{k=0}^{n}{x^{k}}=\frac{1}{1-x}\overbrace{\sum\limits_{k=0}^{n}{\left( x^{k}-x^{k+1} \right)}}^{\text{Telescoping Sum}}=\frac{1-x^{n+1}}{1-x}.\quad\blacksquare
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