how can i proof $3^n$ has a greater growth than $n2^n$?

Question

i want to prove that $3^n$ has a greater growth than $n2^n$ or $3^n = O(n2^n)$. but how can i do it mathematically? by induction or contradiction or other ways of proof.

Answer 1

One simple way would be divide both sides by $2^n$ to get $n$ and $(1.5)^n$. You can simply take a limit of $l i m_{n\to\infty} \frac{(1.5)^n}{n} = \infty$ To show that $(1.5)^n$ grows larger. This limit holds for any number larger than 1.