<s>[INST] Prove that $3^{2n-1} + 2^{n+1}$ is divisible by $7$ for all values of $n$ I have tried to prove this through mathematical induction but I can't seem to prove that the proposition works for $k+1$.
 [/INST] $3^{2n-1}  \begin{matrix} n& 1 & 2 & 3 & 4 & 5 & \cdots \\ mod 7 &3 & 6 & 5 & 3 & 6 & \cdots \end{matrix}$
$2^{n+1} \begin{matrix} n& 1 & 2 & 3 & 4 & 5 & \cdots \\ mod 7 &4 & 1 & 2 & 4 & 1 & \cdots \end{matrix}$
Since $3+4 = 7$, $6+1=7$, $5+2 = 7$ we can see that $3^{2n-1}+2^{n+1} \equiv 0 \pmod{7}$
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