Even simpler…

It’s just a geometric series!

\displaystyle \frac{a^n - 1}{a - 1} = \sum_{i = 0}^{n-1} a^i

Proof

Multiplying:

\displaystyle (a - 1)\sum_{i = 0}^{n-1} a^i = \sum_{i = 0}^{n-1} a^{i+1} - \sum_{i = 0}^{n-1} a^i

Splitting:

\displaystyle (a - 1)\sum_{i = 0}^{n-1} a^i = \sum_{i = 0}^{n-2} a^{i+1} + a^n - a^0 -\sum_{i = 1}^{n-1} a^i

Reindexing:

\displaystyle (a - 1)\sum_{i = 0}^{n-1} a^i = \sum_{i = 1}^{n-1} a^i + a^n - a^0 - \sum_{i = 1}^{n-1} a^i

Cancelling:

\displaystyle (a - 1)\sum_{i = 0}^{n-1} a^i = a^n - 1

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