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Math Help - induction proof

  1. #1
    Junior Member Godfather's Avatar
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    induction proof

    Use Mathematical induction to prove:

    (k^3)=((n^2)(n+1)^2)/4 when the series starts at k=1 and is for n terms
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  2. #2
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    To prove by mathematical induction: first prove for n = 1 ie. substitute n=1 and show that the formula holds for this value.

    Then assume true for n = b: so in this case assume \sum_{k=1}^{b} = \frac{b^2(b+1)^2}{4}

    Then, using the assumption, prove that the formula holds for n=b+1.
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  3. #3
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    Quote Originally Posted by badgerigar View Post
    To prove by mathematical induction: first prove for n = 1 ie. substitute n=1 and show that the formula holds for this value.

    Then assume true for n = b: so in this case assume \sum_{k=1}^{b}  k^3 = \frac{b^2(b+1)^2}{4} Edit by Mr F: Added the forgotten k^3 in the summation.

    Then, using the assumption, prove that the formula holds for n=b+1.
    I'm tipping that the Godfather's problem was the third step of the inductive proof. So here's my help to prevent the mathematics getting murdered:


    \frac{b^2(b+1)^2}{4} + (b+1)^3 = \frac{b^2(b+1)^2 + 4(b+1)^3}{4}


    = \frac{b+1)^2(b^2 + 4[b+1])}{4} = \frac{b+1)^2(b^2 + 4b + 4)}{4} = \frac{(b+1)^2(b+2)^2}{4}


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