Simplifying a geometric series I seem to be completely misunderstanding something about the simplification of a geometric series. $$\sum_{j=1}^{n+1} ar^j = \sum_{j=0}^n ar^j + (ar^{n+1}-a)$$ Why does this work? From what I tested, it doesn't even seem to make sense when you sub in numbers (when I tried $a=3, r=2, n=5$, the first summation yielded $378$, while the second summation yielded $1323$), so what am I misunderstanding?

The sum on the right side contains terms corresponding to $j = 0, 1, ..., n, n + 1$. We then subtract the term for $j = 0$ and the term for $j = n + 1$, so the terms remaining correspond to indices $j = 1, ..., j = n$; this is exactly what the left side represents.