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Math

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n^3+2n is the multiple of 3 prove it by math induction method

  • Math - ,

    step 1:
    test for n = 1
    1^3 + 2(1) = 3 , which is a multiple of 3

    step 2:
    assume it is true for n = k
    that is, k^3 + 3k is a multiple of 3, or it is divisible by 3

    step 3:
    show that it is also true for n = k+1
    that is, show that (k+1)^3 + 3k is a multiple of 3

    let's take the difference
    (k+1)^3 + 2(k+1) - (k^3 + 2k)
    = k^3 + 3k^2 + 3k + 1 + 2k + 2 - k^3 - 2k
    = 3k^2 + 3k + 3
    = 3(k^2 + k + 1)
    which is divisible by 3, (since 3 is a factor)

    so n^3 + 3n is always a multiple of 3

    the property I used is the following:
    if 2 numbers are divisible by the same number, then their difference is divisible by that same number

    e.g. 91 and 49 are both divisible by 7
    then 91-49 or 42 is also divisible by 7
    -- try it for other numbers.
    since we knew the second number, k^3 + 2k , was divisible by 3 and the result was divisible by 3, then the first number, (k+1)^3 + 2(k+1) has to be divisible by 3

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