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Use mathematical induction to prove that 2^(3n) - 3^n is divisible by 5 for all positive integers.

ThankS!

  • Math -

    check for n=1

    2^3 - 3^1 = 8-3 = 5

    assume for k:

    2^(3k) - 3^k = 5m for some m

    now plug in k+1

    2^(3(k+1)) - 3^(k+1)
    = 2^(3k+3) - 3^(k+1)
    = 2^3 * 2^(2k) - 3*3^k
    = 8*2^(3k) - 3*3^k
    = 3*2^(3k) + 5*2^(3k) - 3*3^k
    = 3(2^(3k) - 3^k) + 5*2^(3k)
    = 3(5m) + 5*2^(3k)
    which is a multiple of 5.

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