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Math Help - Maths Induction

  1. #1
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    Maths Induction

    Use Mathematical induction to prove these divisibility results for all positive integers n:

    5^n+2(11^n) is a multiple of 3

    Thanks in advance
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  2. #2
    MHF Contributor alexmahone's Avatar
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    Quote Originally Posted by deltaxray View Post
    Use Mathematical induction to prove these divisibility results for all positive integers n:

    5^n+2(11^n) is a multiple of 3

    Thanks in advance
    P(1):5^1+2(11^1)=5+22=27 is a multiple of 3.

    let P(m) be true.

    5^m+2(11^m)=3k

    2(11^m)=3k-5^m

    P(m+1):5^{m+1}+2(11^{m+1})=5^m.5+2(11^m.11)

    = 5^m.5+(3k-5^m).11

    = 5^m.5+3k.11-5^m.11

    = 33k-6(5^m), which is divisible by 6.

    Thus, P(m+1) is true whenever P(m) is true.

    Hence, by the principle of mathematical induction, 5^n+2(11^n) is a multiple of 3 for all positive integers n.
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