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  "body": "\n## Proof\n\nWe multiply the sum by $1 - r$ and observe that the result telescopes.\n\n1. Let $S_n = \\sum_{k=0}^{n-1} r^k$, the sum of the first $n$ terms of the [[geometric sequence|0MNM54WB99]] $1, r, r^2, \\dots$. For $n = 0$ the sum is empty, so $S_0 = 0$.\n2. Multiply by $r$: $r S_n = \\sum_{k=0}^{n-1} r^{k+1} = \\sum_{k=1}^{n} r^k$.\n3. Subtract step 2 from the definition of $S_n$. Every power $r^k$ with $1 \\le k \\le n-1$ appears once with each sign and cancels.\n4. What remains is $S_n - r S_n = r^0 - r^n = 1 - r^n$. This also holds for $n = 0$, since both sides are $0$.\n5. So $(1 - r) S_n = 1 - r^n$.\n6. Since $r \\ne 1$, the number $1 - r$ is not zero. Dividing step 5 by $1 - r$ gives\n$$\\sum_{k=0}^{n-1} r^k = \\frac{1 - r^n}{1 - r}.$$\n\n## Idea\n\nMultiplying by $r$ shifts every term one place to the right. So the sum and $r$ times the sum share all terms but the first and the last. Subtracting leaves only $1 - r^n$.\n",
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