show by induction that $4^{2n+1}+3^{n+2}$ is divisible by $13$

Question

i'm stuck in this problem, anyone can give me a hint, please?

"Show by induction that $4^{2n+1} + 3^{n+2}$ is divisible by $13$

In my attempt i got the expression

$$4^{2}\cdot 4^{2n+1} + 3\cdot 3^{n+2}$$

but i can't go more, any suggestion is thankful.

$ 4 \cdot 16^n + 9 \cdot 3^n$ and $16 \equiv 3 \pmod {13} $ – Will Jagy Dec 02 '20 at 01:34 — Will Jagy, Dec 02 '20 at 01:34
suggestion: replace $4^2$ with $13+3$ – J. W. Tanner Dec 02 '20 at 01:38 — J. W. Tanner, Dec 02 '20 at 01:38
@J.W.Tanner excelent! Thanks. – Davis M Dec 02 '20 at 01:59 — Davis M, Dec 02 '20 at 01:59

score 5 · Accepted Answer · answered Dec 02 '20 at 01:35

5

\begin{eqnarray*} 4^{2}\cdot 4^{2n+1} + 3\cdot 3^{n+2}= 16(4^{2n+1}+3^{n+2}) -13 \times 3^{n+2}. \end{eqnarray*}

answered Dec 02 '20 at 01:35

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