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Use mathematical induction to prove the property for all positive integers n.

A factor of (n^3+3n^2+2n) is 3.

  • Math -

    1^3+3*1^2+2*1 = 1+3+2 = 6 = 3*2
    so, true for n=1

    so, assuming it's true for n=k, work with n=k+1:

    (k+1)^3 + 3(k+1)^2 + 2(k+1)
    = k^3+3k^2+3k+1 + 3k^2+6k+3 + 2k+2
    = k^3+3k^2+2k + 3k+1 + 3k^2+6k+3 + 2
    we know that k^3+3k^2+2k is a multiple of 3, so check out the rest. Rearranging things, we have

    3k^2+9k+3

    This is clearly also a multiple of 3.

    So,
    P(1)
    P(k) => P(k+1)
    so P is true for all k.

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