If $a$ and $p$ are relatively coprime integers, then is there any efficient way of calculating the following: $(\sum_{k=0}^n\frac{1}{a^k}) \% p$ ? I'm interested in the cases when $0 \leq n ≤ 2147483647$ and $2 ≤ p ≤ 2147483647$. Thanks.
finding the sum of series
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sequences-and-series