Understanding Less Frequent Form of Induction? (Putnam and Beyond) I won't paste the question here since my problem is not a technical one but a conceptual one. 
Book is here: (Page 22 of the pdf)
I do not understand why it is necessarily to induct $2^{k}$ to show $2^{k+1}$ when the proof follows up with inducting backwards from (n+1) to (n). 
I recall a similar induction problem where backwards induction itself already suffice.
Why it is necessarily to show the former $2^{k}$? Doesn't showing (n+1) to (n) already include in-itself the powers of $2$?

Please do insert more details in your question.
In any case, forward-backward induction is a very common concept, so here's my answer:
It is not sufficient to be able go from $n+1$ to $n$ because you don't have a large value of $n$ to start.
The forward part of going from $2^k$ to $2^{k+1}$ ensures that you have arbitrarily large values of $n$ from which you can go downwards.
