Prove $5^n +5 <5^{n+1}$ $∀n∈N$ using induction Prove $5^n +5 <5^{n+1}$ $∀n∈N$
Base Case: $n=1$
$\implies 5^1 +5 <25$
$\implies 10<25$ ; holds true
Induction hypothesis: Suppose $5^k +5 < 5^{k+1}$ is true for k∈N
Then;
$\implies 5^{k+1} +5 < 5^{k+2}$ 
$\implies 5\cdot 5^k +5 < 25*5^k$
I don't know how to proceed after this step.

You don't really need induction:
$$
5^n + 5 \le 5^n + 5^n = 2\cdot 5^n < 5\cdot 5^n = 5^{n+1}
$$
but you can justify $5 \le 5^n$ by induction if you must.
