Stuck on Mathematical Induction Proof I have the following question: Prove with mathematical induction that $3^n+4^n\le 5^n$ for all $n\ge 2$.
$$\text{Assume true: }3^n+4^n \le 5^n \text{. Prove that $3^{n+1}+4^{n+1} \le 5^{n+1}$} \\
= 3\cdot3^n+4\cdot4^n \\ =3\cdot3^n+4^n(3+1)\\
=3\cdot3^n+3\cdot4^n+4^n \\ 
=3(3^n+4^n)+4^n \\
\le 3(5^n)+4^n \\
(5-2)(5^n)+4^n \\
5^{n+1} - 2\cdot5^n+4^n\\
5^{n+1} - 2\cdot(3^n+4^n) +4^n$$
I am stuck on. I see now that I double the $5^n$, and this leads me nowhere. Where can I go from here to solve this? I have tried playing with the algebra, but I am not getting anywhere with this last step.

From where you left off, we have
$$
5^{n+1}-2(3^n+4^n)+4^n=5^{n+1}-2\cdot3^n - 4^n\leq 5^{n+1}
$$
and your proof is finished.
