# Math Help - Proofs

1. ## 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.

2. Hello,
Originally Posted by dlee426
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.

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}
2^{2k+3}+3^{2k+3}
&=4 \cdot 2^{2k+1}+9 \cdot 3^{2k+1} \\
&=4 \cdot 2^{2k+1}+9 \cdot (5n-2^{2k+1}) \\
&=4 \cdot 2^{2k+1}+5 \cdot (9n)-9 \cdot 2^{2k+1} \\
&=2^{2k+1} (4-9)+5 \cdot (9n) \end{aligned}

...

3. 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.