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Sum of the fourth powers of the first n positive integers

  1. Jun 30, 2009 #1
    1. The problem statement, all variables and given/known data

    Find a formula fo the sum of the fourth powers of the first n positive integers

    n
    ∑ i^4
    (i=1)


    Justify your work using mathematical induction

    2. Relevant equations

    so i know the formula for the sum of the cubes of the first n positive integers

    k=n+1
    ∑ = (1^3)+(2^3)+(3^3)+....+(n^3)+((n+1)^3)= {((n+1)^2)((n+2)^2)} / (4)
    k=1

    I was wondering what was the proof for the sum of the quartic of the first n positive integers

    3. The attempt at a solution

    This is actually what I started working out and I dont know whether it is right

    N
    ∑ i^4 = (1/30)(N+1)(N)(2N+1)((3N^2)+3N-1)
    i=1
     
    Last edited: Jun 30, 2009
  2. jcsd
  3. Jun 30, 2009 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Sure, that's right. I know it's right because I looked it up. Just like you, probably. The problem is that you have to prove it's right. Call your sum S(N). Then the inductive step (after you shown it's true for N=1) is to show S(N+1)-S(N)=(N+1)^4. Do you see why? If you see why, that's the important part.
     
    Last edited: Jun 30, 2009
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