I'm guessing you mean

$4^{n+1}+5^{n-1}$ is divisible by $21, ~~\forall n \in \mathbb{N}, n\geq 1$

But maybe not since

$n=1 \Rightarrow 4^2+1=17 \neq 21k$

$n=2 \Rightarrow 4^3+5 = 69 \neq 21k$

I show that in general your statement is incorrect.

Could you repost the correct statement you're trying to prove?