I'm working through Spivak's Calculus (Fourth Edition).
In chapter 2, problem 6:
The formula for $1^2 + ... + n^2$ may be derived as follows. We begin with the formula $(k+1)^3 - k^3 = 3k^2 +3k +1$
Writing this formula for $k = 1, ... , n$ and adding, we obtain
$(n+1)^3 - 1= 3 (1^2 + ... + n^2 ) +3(1+...+n) +n$
Thus we can find $(1^2 + ... + n^2 )$ if we already know $(1+...+n)$ (which could have been found in a similar way).
This is clear for me. However, I'm stuck in this problem: Use this method to find $$\frac{3}{1^2\times2^2} + \frac{5}{2^2\times3^2} +\cdots+ \frac{(2n+1)}{n^2(n+1)^2}$$
I would like to know how could the one think in this and not just how to solve it. I want to know how to deal with "deriving summation formulas" in general.