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Math Help - Mathematical induction proof

  1. #1
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    Mathematical induction proof

    Hello, I was wondering if anyone can help me with this problem.

    Find a closed form for
    /sum_{k=0}^n x^k where x does not equal to zero.

    Then prove that it is valid for all integers n>= 1.
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  2. #2
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    Quote Originally Posted by Recklessid View Post
    Hello, I was wondering if anyone can help me with this problem.

    Find a closed form for
    /sum_{k=0}^n x^k where x does not equal to zero.

    Then prove that it is valid for all integers n>= 1.
    This is the geometric sum:
    1+x+x^2+...+x^n
    Let S=1+x+...+x^n
    Then,
    xS=x+x^2+...+x^{n+1}
    Thus,
    S-xS=(1+x+...+x^n)-(x+x^2+...+x^{n+1})=1-x^{n+1}
    S(1-x)=1-x^{n+1}
    Now if x is not equal to 1 then,

    S=(1-x^{n+1})/(1-x)

    And if x is equal to 1 then,
    1+x+x^2+...+x^n=1+1+...+1=n

    Thus,
    sum[k=0,n]x^k = (1-x^{n+1})/(1-x) if x not = 1, n if x =1.
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