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Math Help - Proofs

  1. #1
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    Proofs

    I need to show that 2^n + 3^n is a multiple of 5 where n is an odd number using mathematical induction.

    I am pretty far along the problem, having done the induction, calling m=2k+1, then plugging in m+2=2k+3, but I cannot manipulate the problem to show that it is a multiple of 5.

    Thanks in advance for your help.
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  2. #2
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    Hello,
    Quote Originally Posted by dlee426 View Post
    I need to show that 2^n + 3^n is a multiple of 5 where n is an odd number using mathematical induction.

    I am pretty far along the problem, having done the induction, calling m=2k+1, then plugging in m+2=2k+3, but I cannot manipulate the problem to show that it is a multiple of 5.

    Thanks in advance for your help.
    So you want to prove that 2^{2k+3}+3^{2k+3} is a multiple of 5, knowing that 2^{2k+1}+3^{2k+1}=5n, for some integer n (inductive hypothesis)
    Note that 3^{2k+1}=5n-2^{2k+1}

    Hence :
    \begin{aligned}<br />
2^{2k+3}+3^{2k+3}<br />
&=4 \cdot 2^{2k+1}+9 \cdot 3^{2k+1} \\<br />
&=4 \cdot 2^{2k+1}+9 \cdot (5n-2^{2k+1}) \\<br />
&=4 \cdot 2^{2k+1}+5 \cdot (9n)-9 \cdot 2^{2k+1} \\<br />
&=2^{2k+1} (4-9)+5 \cdot (9n) \end{aligned}

    ...
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  3. #3
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    Without induction, consider that 3 \equiv -2 \mod 5, so 2^n+3^n \equiv 2^n+(-2)^n \mod 5. For odd values of n, you can factor out the negative, giving what you want.
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