Product of Sequence Terms (Jee Mains)

Arithmetic and Geometric Progression

Hard

Product of Sequence Terms

Let 729, 81, 9, 1, …. be a sequence and P_n denote the product of the first n terms of this sequence.
If $2{\sum\limits_{n = 1}^{40} {\left( {{P_n}} \right)} ^{\frac{1}{n}}} = \frac{{{3^\alpha } - 1}}{{{3^\beta }}}$ and gcd (α, β) = 1, then α + β is equal to

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