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Given a series starts with $N$ positive values and the $i + 1$ (for $i + 1 > N$) value of the series is defined as the average of the last $N$ values. $$x_{i+1}=\frac{1}{N}\sum_{k=i-N+1}^i x_k$$

Is it possible to find a closed formula for this sequence in terms of the first N values?

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    This is a [linear recurrence](https://en.wikipedia.org/wiki/Constant-recursive_sequence), so yes, you may derive a closed formula (much like Binet's formula for Fibonacci numbers). It won't be nice, though. – 2017-01-26
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    You can use the characteristic polynomial method of solving recurrence relations to get a characteristic polynomial of $Nr^{N+1} - (N+1)r^N + 1 = 0$, but that doesn't look easily factorable. – 2017-01-26

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