<s>[INST] Proving that $5^n - 1$ is divisible by $4$ by mathematical induction. I have done it, but I am not sure that the inductive step is right. Can anybody please clear me about it? 
Basic steps as:
Taking $n=1$: $p(1)=5-1=4$.
Inductive hypothesis:  Assume the statement is true for $p(k)$.  $5^k - 1$ is divisible by $4$.
Inductive steps: We must show $p(k+1)$ is true when $p(k)$ is true.
\begin{align*}
& 5^k -1 + 5^{k+1} -1\\
& 5^k -1 + 5.5^{k} -1\\
& (5^k -1) + 4
\end{align*}
 [/INST] *

*For n=1 the statement is true.

*Assume that $$5^n-1$$ is divisible by 4.

*Let us consider $$5^{(n+1)}-1$$. We have to prove that it is divisible by4 also.


We have that
$$
\begin{gathered}
  5^{n + 1}  - 1 = 5 \cdot 5^n  - 1 = 5 \cdot 5^n  - 5 + 5 - 1 =  \hfill \\
   \hfill \\
   = 5\left( {5^n  - 1} \right) + 4 \hfill \\ 
\end{gathered} 
$$
Since by inductive step 2 you have that $$5^n-1=4k$$ whit k integer you have that
$$
5\left( {5^n  - 1} \right) + 4 = 5 \cdot 4k + 4 = 4\left( {5k + 1} \right)
$$
and this prove step 3
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