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prove by mathematical induction that 7n+4n+1 is divisible by 6

  • maths - error - ,

    As written, it's obviously false. Try n=2
    Did you mean 7n^2+4n+1? Nope; false for n=2

    Got some error here

  • maths - ,

    its 7^n+4^n+1

  • maths - ,

    Steve

  • maths - ,

    Ah; that's a lot nicer.

    it's true for n=1.
    So, assume it's true for n=k.

    7^(k+1) + 4^(k+1) + 1
    = 7*7^k + 4*4^k + 1
    = (1+6)*7^k + (1+3)*4^k + 1
    = (7^k+4^k+1) + 6*7^k + 3*4^k

    obviously, 6*7^k is divisible by 6
    4^k is even, so 3*4^k is divisible by 6
    So, since we're adding three items which are all multiples of 6, the whole is a multiple of 6.

    Thus, the induction step holds.

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