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Math Help - Prove divisibility by induction

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    Prove divisibility by induction

    Prove that n^3+2n is divisible by 3.
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    Quote Originally Posted by cityismine View Post
    Prove that n^3+2n is divisible by 3.
    We're going to try this with LaTeX and hope for the best.

    Case 1: n = 1
    1^3 + 2 \cdot 1 = 1 + 2 = 3
    Check!

    Now assume the theorem is true for n = k. We need to show it is true for n = k + 1. So we assume
    k^3 + 2k = 3a
    where a is an integer.

    Thus we need to show the same for
    (k + 1)^3 + 2(k + 1)

    = k^3 + 3k^2 + 3k + 1 + 2k + 2

    = (k^3 + 2k) + (3k^2 + 3k + 3)

    The first term is divisible by 3 due to our assumption at n = k. The second term is obviously divisible by 3.

    etc, etc.

    -Dan
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  3. #3
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    That sort of makes sense to me, but I still don't understand induction very well. It's the last chapter in my pre-calculus textbook. Do I need to learn it well for 1st year calculus?
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