There is a question in the book that I don't quite understand.
Question Show that $n^2$ is smaller than $2^n$ whenever $n\ge5$.
At the $k+1$ step it gets very whacked and confusing.
$k+1$ step
$$2^{k+1} = 2 \cdot 2^k > k^2 + k^2 > \mathbf{k^2 + 4k > k^2+2k+1}$$
Can someone explain to me where the book magically pulled out $4n$ because it seems to me that they replaced one of the $k^2$ by $4k$, but I have no idea why they did it and where that number $4k$ came from.
If you leave out the $4n$-step, you only have to see that $$k^2 > 2k+1 \qquad\forall\ k>5$$ This is rather simple to see since $2k+1 < 3k < k\cdot k = k^2$
It occurs to me that that $4n$ was supposed to say $4k$ wich is fine and works just as well as the $3k$-term does in my alternative.