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Math Help - Prove Binomial Theory

  1. #1
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    Prove Binomial Theory

    Prove Binomial Theory by Mathematic Induction
    Cn,r<(n+1)^r ;0<=r<=n
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  2. #2
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    Quote Originally Posted by calfever View Post
    Prove Binomial Theory by Mathematic Induction
    Cn,r<(n+1)^r ;0<=r<=n
    Use Google.

    proof binomial theorem induction - Google Search=
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    Quote Originally Posted by calfever View Post
    Prove Binomial Theory by Mathematic Induction
    Cn,r<(n+1)^r ;0<=r<=n
    I think Mr. F has misinterpreted the problem statement due to its misleading title. I think the problem is:

    Prove that
    C(n,r) \leq (n+1)^r for all 0 \leq r \leq n ....(*)
    by mathematical induction.
    Note that I have changed the less-than in the original statement to less-than-or-equal. Otherwise (*) would be false when r=0.

    I'm going to assume you know the important identity C(n,r-1) + C(n,r) = C(n+1,r).

    First check the case n = 0:
    If n=0 then the only possible value of r is r=0.
    C(0,0) = 1 and (0+1)^0 = 1, so (*) is true.

    Now suppose that for some integer k we have
    C(k,r) \leq (k+1)^r for all 0 \leq r \leq k.
    I will let you check the case r=0.
    If r \neq 0 then 0 \leq r-1 \leq k, so by assumption we also have
    C(k,r-1) \leq (k+1)^{r-1}.

    Adding the two inequalities,
    C(k,r) + C(k,r-1) \leq (k+1)^r + (k+1)^{r-1}
    so by the identity stated above,
    C(k+1,r) \leq (k+1)^{r-1} (k + 1 + 1) < (k+2)^{r-1} (k+2) = (k+2)^r
    so (*) holds for n=k+1

    By mathematical induction, (*) is true for all n \geq 0.
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  4. #4
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    thanks a lot!! AWKWARD
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