<s>[INST] Prove by mathematical induction that Prove by mathematical induction that 
$$P(n)=3^{2n+1} + 2^{n-1}$$
is a multiple of 7.
My Attempt:
Here, $P(n)= 3^{2n+1}+2^{n-1}$
For $n=1$,
$$P(1)=3^3+2^0$$
$$=28$$
So, $P(1)$ is true. 
Suppose, $P(m)$ is true for all $m\in N$
Now, we have to show that $P(m+1)$ is true,
$$P(m+1)=3^{2(m+1)+1}+2^{(m+1)-1}$$
$$=3^{2m+3}+2^m$$
 [/INST] Inductive Step
$$
\begin{align}
P(m+1)
&=3^{2m+3}+2^m\\
&=9\cdot3^{2m+1}+2\cdot2^{m-1}\\
&=7\cdot3^{2m+1}+2\left(3^{2m+1}+2^{m-1}\right)\\
&=7\cdot3^{2m+1}+2P(m)
\end{align}
$$

Direct Proof
Note that since $2\cdot4\equiv1\pmod7$, we get
$$
\begin{align}
3^{2k+1}+2^{k-1}
&=3\cdot9^k+2^{-1}\cdot2^k\\
&\equiv3\cdot2^k+4\cdot2^k&\pmod7\\
&=7\cdot2^k\\
&\equiv0&\pmod7
\end{align}
$$
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