<s>[INST] Prove by induction $6\cdot 7^n -2\cdot 3^n$ is divisible by $4$ for $n\ge 1$ Prove by induction $6\cdot 7^n -2\cdot 3^n$ is divisible by $4$ for $n\ge 1$.
I am struggling on this problem very much.
So far I have Basis case = $6\cdot 7^1 - 2\cdot3^1 = 36$ which is divisible by $4$
Assume for a $n$ that $6\cdot 7^n-2\cdot3^n$ is divisible by $4$.
After that I am not sure what to do.
 [/INST] Let $P(n) = 6\cdot 7^n - 2\cdot 3^n$. If $4| P(1)$ and we assume $P(k)$ to be true for some arbitrary natural number $k$ and we can show $4| P(k+1) - P(k)$ then we have proved that $4|P(n) \ \forall n\in\mathbb{N}$.
$$\begin{align*} P(k+1) - P(k) & =  6\cdot 7^{k+1} - 2\cdot 3^{k+1} - 6\cdot 7^k + 2\cdot 3^k \\ & = 7\cdot 6\cdot 7^k - 3\cdot 2\cdot 3^k - 6 \cdot 7^k + 2\cdot 3^k \\ & = 6\cdot 6 \cdot 7^k - 2 \cdot 2 \cdot 3^k \\ & = 4 \cdot (9\cdot 7^k - 3^k)\end{align*}$$
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