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Finite Differences proving

  1. Aug 9, 2005 #1
    I am to prove by mathematical induction that for a polynomial of degree n p(x) with leading coefiicient a_0,

    \Delta^n p(x) = a_o n!

    My proof: By mathematical induction

    \Delta^1 p(x) = [a_0(x+1) + a_1] - [a_0x + a_1]
    \Delta^1 p(x) = [a_0x + a_0 + a_1] - [a_0x + a_1]
    \Delta^1 p(x) = a_0
    \Delta^1 p(x) = a_0 \cdot 1!

    hence, S(1) is true

    This is where I have a problem. I assume that [tex]\Delta^n p(x) = a_o n! [/tex] is true... how do i show that S(n+1) is also true? The degree of the polynomial becomes n+1 and my S(n) becomes inapplicable already...
  2. jcsd
  3. Aug 10, 2005 #2


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    Staff Emeritus
    Science Advisor

    What is the first difference of a polynomial of degree n+1? Isn't it a polynomial of degree n? What is its leading coefficient?
  4. Aug 10, 2005 #3


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    Homework Helper

    First show that n=1,2,...
    [tex]\Delta x^{n}=n x^{n-1} +P_{n-2}[/tex]
    where P_{n} means some polynomial of degree at most n
    and thus P_0=0
    [tex]\Delta P_{0}=0[/tex]
    then show by induction that for k=0,...,n
    [tex]\Delta^k x^n=\frac{n!}{(n-k)!}x^{n-k}+P_{n-1-k}[/tex]
    Take your result as a special case
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