My question is simple. Could someone explain in the simplest manner possible, how to find a closed form of a summation in general? Please explain it step by step because the book spends literally two sentences discussing the process.
Take this as an example problem. This is as far as I've gotten. Other articles seem to suggest that there is usually a pattern, and there's no standardized technique to find things like that.
HINT:
$$\left(3+\dfrac{2r}n\right)^2=9+\dfrac{12}n\cdot r+\dfrac4{n^2}\cdot r^2$$
$$\sum_{r=1}^n1=n$$
$$\sum_{r=1}^nr=\dfrac{n(n+1)}2$$
$$\sum_{r=1}^nr^2=\dfrac{n(n+1)(2n+1)}6$$