TITLE: $3^n+4^n<5^n$ for all $n>2$
QUESTION [8 upvotes]: I'm doing the following induction proof and wanted to know if this was valid. I think it is, but I'm seeing more complicated solutions than what I did. What I did seems much easier.
Prove that $3^n+4^n<5^n$ for all $n>2$.
When $n=3$ we get $91<125$. No problem, now assume the result is true from $k<n$, $(5^k>3^k+4^k)$ and consider $5^{k+1}=5 \times 5^k>5(3^k+4^k)=5\times 3^k + 5\times 4^k>3\times 3^k+4\times 4^k=3^{k+1}+4^{k+1}$ since $5\times 3^k>3\times 3^k$ and $5\times 4^k> 4\times 4^k$.

REPLY [3 votes]: Another proof, NO INDUCTION! -- just for fun.
Let $\ n\ge 3.\ $ then:
$$ 3^n + 4^n <\, 3^2\cdot 5^{n-2} + 4^2\cdot 5^{n-2}
 \,=\, (3^2+4^2)\cdot 5^{n-2}\, =\ 5^n $$


REMARK   If you look at the above proof under a magnifying glass then you may see induction after all -- just a trace.