Is $2^n + n < 2^{(n+1)}$?

Question

Is $ 2^n + n < 2^{(n+1)}$ when $ n > 2 $

How can I prove this? I need it in order to prove a language is context free

I'm voting to close as off-topic because this is a purely mathematical question with no computational content. — David Richerby, May 10 '18 at 14:49

score 0 · Answer 1 · answered May 10 '18 at 14:41

0

Write $2^{n+1}$ as $2^n\cdot2$. Now, prove that $2^{n+1}-2^n>n$.

answered May 10 '18 at 14:41

zdm

score 0 · Answer 2 · answered May 10 '18 at 15:47

0

As $2^{n+1} = 2^n*2 = 2^n + 2^n$ so we just need to prove $2^n > n$.

We can do that by induction really easily.

For $n = 2$ then $2^n = 2^2 = 4 > 2 = n$.

And if $2^n > n$ then

$2^n + 1 > n + 1$ and

$2^{n+1} = 2*2^n = 2^n + 2^n > 2^n + 1 > n+1$.

answered May 10 '18 at 15:47

fleablood

2 Answers2