Geometric Sequence. How to check if a given  sum  represents the sum of the terms of a geometric sequence?

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The sum of n terms of a geometric sequence is given as a*(r^n - 1)/(r - 1), where r is the common ratio and a is the first term

4^n - 1

=> (2^(2n) - 1) / (2 - 1)

Here a = 1 and r = 2. But 2n can only take on even values.

The expression 4^n - 1 can be the sum of terms of a geometric sequence but it holds only when the number of terms is even.

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