In a book I have they solve an equation like this:
$$\sum_{i=2}^{19} i(i - 2) = \sum_{i=2 - 1}^{19 - 1} (i + 1)(i + 1 - 2) = \sum_{i=1}^{18} (i + 1)(i - 1) = \cdots$$
What I don't get is how it works. If I do $-1$ on the summation index, why do I have to add $+1$ to the variable $i$ used in the terms? Is there a name for this technique and the rules associated with it?
This has no special name, it is just "shifting of the summation index". I think it works best for beginners if you give the shifted index a new name. Let us start with the first sum and then say you want a new summation index
$$ k = i - 1 .$$
If you sum $i$ from 2 to 19, that means you sum $k$ from $1= 2 -1 $ to $18 = 19 - 1$. And to see what happens to the summands, compute as follows, useing that $k=i-1$,
$$\sum_{i=2}^{19} i (i-2) = \sum_{i=2}^{19} (i-1+1)(i-1-1)= \sum_{k=1}^{18} (k+1)(k-1). $$
So you see that the "+1" in the summands compensates the shift down in the summation index. And now you can rename $k$ to $i$ or anything else if you like, because the name of the summation index does not matter at all (it is a so-called "silent" index).
If you adjust your clock such that the time it shows is one hour earlier than the actual time (e.g. showing $1$ o'clock when it is actually $2$ o'clock), then you can arrive your appointment one hour later according to the adjusted clock, and still be on time.
