Proof by induction: $5^n \geq 5n^3 + 2$ for $n \geq 4$ One of the practice problems I have is to prove by induction that for every $n \geq 4$ the following inequality holds:
$5^n \geq 5n^3 + 2$
My progress so far (inequality holds for base case $n=4$):
$5^{n+1} \geq 5(5n^3 + 2)$
$5^{n+1} \geq 25n^3 + 10$
The next step logical step for me is to prove that  $25n^3 + 10 \geq 5(k+1)^3 + 2$ but I have no idea how.

You have $5^{n+1}=5\cdot 5^n\geq 5(5n^3+2)=25n^3+10$, we need to show that
$5^{n+1}\geq 5(n+1)^3+2=5(n^3+3n^2+3n+1)+2=(5n^3+15n^2+15n+5)+2$
We know that $n\geq 4$, so if we write:
$25n^3=5n^3+20n^3\geq 5n^3+80n^2\geq 5n^3+15n^2+65n^2\geq 5n^3+15n^2+260n$
$\geq 5n^3+15n^2+15n+5=5(n+1)^3$
Where we substitute one $n$ by $4$ in every step and derive the result like that.
So: $25n^3+2\geq 5(n+1)^3+2$, which ends the inductive proof.
