<s>[INST] Proving that $5^n-1$ is divisible by $4$ for $n\geq 0$ by induction I hope this is not counted as a duplicate, as I would like to know if my proof is valid:
$P(n): 5^n - 1$ is divisible by $4$ for $n \ge 0$.
Base Step: $P(0): 5^0-1 = 1-1 = 0 = 0\times 4$.
Induction Supposition: $P(k): 5^k-1$ is divisible by $4$.
Prove: $P(k+1): 5^{k+1}-1$ is divisible by $4,$ or equivalently $5^{k+1}-1 = 4r$, for some integer $r$.
$5^{k+1}-1$
$= 5^k\times5-1$ by Exponent Laws
$= 5\times4r$ by I.H.
$=4(5r)$ which was to be shown.
 [/INST] For $n=1$ we have $5^{n} - 1 = 5 - 1 = 4.$
Suppose there is an $n \geq 1$ such that $5^{n} - 1 = 4k$ for some $k$. Then
$$5^{n+1} - 1 = 5(5^{n} - 1) + 4 = 20k + 4,$$
so
$$4 \mid (5^{n+1} - 1).$$
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