Induction with inequality problem Prove by induction that $2k(k+1) + 1 < 2^{k+1} - 1$ for $ k  > 4$.
Can some one pls help me with this?
I reformulated like this
$ 2k(k+1) + 1 < 2^{k+1} - 1 $
$     2k^2+2k+2<2^{k+1}$
and I tried like this
Take $k=k+1$
$ 2^{k+2} -1 >  2(k+1)(k+2) + 1  $
$2^{k+2} >   2(k+1)(k+2) + 2$
$   2^{k+2} >      2k^2+2k+2  +4k+4$
I dont know how to proceed further
Please help me.

You want to show $2k(k+1)+2<2^{k+1}$
For this, $k=5$ holds true (why?)
Further, $2^{k+2} = 2\cdot 2^{k+1} > 2\cdot(2k^2+2k+2) = 4k^2 + 4k + 4 = (2k^2 + 6k + 6)+2(k^2-k-1) $
Now can you recognise the first part in brackets and show that the second part is positive for $k> 4$?
