Prove by induction that (5^(n))-1 is divisible by 4 for all natural numbers n.

Question

Prove by induction that $5^n-1$ is divisible by $4$ for all natural numbers $n$.

I got $P(k+1)=5^{k+1}-1$ but I don't where to go now.

Answer 1

Note that $5^{k+1}-1=5(5^k-1)+4$. Now use your induction assumption that says $4|5^k-1$.

Answer 2

$5^{k+1}-1=(5^{k}-1)\cdot 5+4$

If $5^{k}-1$ is divisible by 4, what can you say about this expression?

Answer 3

Although induction is required, perhaps you might also be interested in a non-inductive proof. Since $5 \equiv 1 \pmod{4}$, then $5^k \equiv 1 \pmod{4}$ for all $k \in \mathbb{N}$.

And so we have $5^n - 1 \equiv (1-1) \equiv 0 \pmod{4}$. We conclude that $4|(5^n - 1)$ for all $n \in \mathbb{N}$.