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Math Help - Induction and Recursion

  1. #1
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    Induction and Recursion

    Problem :
    Prove that for every positive integer n,
    1 * 2 * 3 + 2 * 3 * 4 + . . . . . + n(n+1)(n+2) = n(n+1)(n+2)(n+3)/4

    Question :
    Basis Step:
    I want to prove this by determining if P(0) is true which in this case 0(0+1)(0+2)(0+3) = 0 .

    Induction Step:
    1 * 2 * 3 + 2 * 3 * 4 + . . . . . + k(k+1)(k+2) + (2k+1) =
    [k(k+1)(k+2)(k+3)/4 + (2k+1)]

    When I simplify I cannot come up with the original solution n(n+1)(n+2)(n+3)/4. I believe I am doing something wrong trying to prove the induction. Any help is appreciated!
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  2. #2
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    Quote Originally Posted by Ramin_Shahab View Post
    Problem :
    Prove that for every positive integer n,
    1 * 2 * 3 + 2 * 3 * 4 + . . . . . + n(n+1)(n+2) = n(n+1)(n+2)(n+3)/4

    Question :
    Basis Step:
    I want to prove this by determining if P(0) is true which in this case 0(0+1)(0+2)(0+3) = 0 .

    Induction Step:
    1 * 2 * 3 + 2 * 3 * 4 + . . . . . + k(k+1)(k+2) + (2k+1) =
    [k(k+1)(k+2)(k+3)/4 + (2k+1)]

    When I simplify I cannot come up with the original solution n(n+1)(n+2)(n+3)/4. I believe I am doing something wrong trying to prove the induction. Any help is appreciated!
    Step 2: Assume true for n = k:

    1 * 2 * 3 + 2 * 3 * 4 + . . . . . + k(k+1)(k+2) = k(k+1)(k+2)(k+3)/4


    Step 3: Show true for k = n+1.

    1 * 2 * 3 + 2 * 3 * 4 + . . . . . + k(k+1)(k+2) + (k+1)(k+2)(k+3) = k(k+1)(k+2)(k+3)/4 + (k+1)(k+2)(k+3)

    using step 2.

    And it's trivial to show that k(k+1)(k+2)(k+3)/4 + (k+1)(k+2)(k+3) = (k+1)(k+2)(k+3)(k+4)/4 .....
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