Q:

Simple induction proof that $3^n + 5^n \ge 2^{n+2}$

Im having a lot of trouble proving by induction that $3^n + 5^n \geq 2^{n+2}.$
The base step is easy, but I don't seem to find the way to proof the inductive step.

A:

Inductive step
$$3^{n+1} + 5^{n+1}=3\times3^{n} + 5\times5^{n}\geq2(3^{n} + 5^{n}) \geq2\times 2^{n+2}=2^{n+3}$$

A:

Hint $\ $ Dividing by $2^n$ it is $(3/2)^n + (5/2)^n \ge 4.\,$ But for any $\,a,b\ge 1,$ $\,f_n = a^n+b^n\,$ is increasing, therefore $\,f_1\ge 4\,\Rightarrow\, f_n\ge f_1\ge 4.\,$ 
Remark $\ $ By a  little rearranging, we reduced the induction to more intuitive inductions about increasing functions (which still require inductive proof, but the proofs are straightforward, using prior intuition about increasing functions).  In the same way, such intuitive preprocessing often allows one to transform inductive proofs into simpler or well-known forms, e.g. see many prior posts on induction via telescopy.

