I found this equation in a computer science article:
$$ \sum^{m}\_{i=0} N^{i} = \frac{N^{m+1}-1}{N-1} $$
and I'm really unsure why this is true. I tried to come up with an answer myself, I arbitrarily chose $m$ to be equal to 4, here are my equations:
$$ \sum^{4}\_{i=0} N^{i} = 1 + N + N^{2} + N^{3} + N^{4} = N(N^{3} + N^{2} + N + 1) + 1 $$
After this I'm not sure where to go, somehow I should be able to reach $N^{5}$ (as in $\frac{N^{5}-1}{N-1}$) but I don't see how that's possible.