Cannot understand how the last two equalities are obtained. Could someone help?
Note that $\Large\sum\limits_{k = 1}^N {f(k)} - \sum\limits_{k = 1}^N {f(k - 1)} = f(N) - f(0)$. If that is not clear then write out the sum with $N=5$. If it is still not clear, see your tutor.
Oh okay. So we offset the two sums by one step and hence get the last and first one. I was thinking about direct transformation of the sums into their corresponding terms, hence couldn't understand what happened.