The N-series (edited)

Number Theory Level pending

Let \( \displaystyle N_{k} (a) = \sum_{i=0}^k a^{i} \) for any positive odd integer \(a\).

For example, \( N_{5} (a) = 1 + a + a^{2} + a^{3} + a^{4} + a^{5} \).

What is the largest power of 2 that can completely divide all numbers of the form

\[ \large (N_{95}+ N_{63} - N_{31})(a) ?\]

Clarifications:

\(a\) has to be an odd integer.

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