How do I prove the following inquation using induction?
$2^n>1+n\sqrt{2^{n-1}}$ , $n\geq 2$
I did the base case, but I'm stuck at the induction process.
The induction: $2^{k+1}>1+(k+1)\sqrt{2^{k}}$
Now I used the hipothesis to prove the induction:
$2*2^{k}>2*(1+k\sqrt{2^{k-1}})$
$2^{k+1}>2+2k\sqrt{2^{k-1}})$
I can´t do the rest
Hip: $2^n>1+n\sqrt{2^{n-1}}$.
Then,
$2 \cdot 2^n>2(1+n\sqrt{2^{n-1}})$
$2^{n+1}>2+2n\sqrt{2^{n-1}}=2+n\sqrt{2}\sqrt{2^{n}}>1+(n+1)\sqrt{2^{n}}$
since $n\sqrt{2}>n+1$, if $n>2$.
Shouldn't your induction hypothesis be $2^k > 1 + k\sqrt{2^{k-1}}$ for some integer $k \geq 2$? It looks like you assumed $2^k > 1 + k\sqrt{2^{k}}$ instead.
