Sequences & Series · Geometric progression sum

Let

729, 81, 9, 1, \ldots

be a sequence and

P_n

denote the product of the first

n

terms of this sequence. If

2\sum_{n=1}^{40}\left(P_n\right)^{\frac1n} = \dfrac{3^{\alpha} - 1}{3^{\beta}}

and

\gcd(\alpha, \beta) = 1

, then

\alpha + \beta

is equal to

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