Solving Recursive Equations - Domain Transformation

Hi all,

So I've solved a recursive equation so far and I'm about to prove that it is correct through induction.

However, before I get into that... here's what I have:

http://i.imgur.com/nNNW9vH.png

I have 2 questions:

(1) Why, at the end of the telescoping, does S(k) - 1 = k?

How am I supposed to know what the final equation will equal to?

(2) Why, before the end of the telescoping, do the equations = 1?

How am I supposed to know what the equations are supposed to add up to?

I asked my professor for those two parts, he gave me the answers (1 and k) and explained why, but I can't remember the reasoning.

I figure that if I understand the answers to my 2 questions, I will be able to solve more difficult recursive equations more easily.