Let 729,81,9,1, … be a sequence and P_n denote the product of the first n terms of this sequence. If 2 Σ_n=1^40( P_n)^(1/n)=(3^α-1)/(3^β) and gcd (α,…

Let $\displaystyle 729,81,9,1, \ldots$ be a sequence and $\displaystyle \mathrm{P}_n$ denote the product of the first $\displaystyle n$ terms of this sequence.If $\displaystyle 2 \sum_{n=1}^{40}\left(\mathrm{P}_n\right)^{\frac{1}{n}}=\frac{3^\alpha-1}{3^\beta}$ and $\displaystyle \operatorname{gcd}(\alpha, \beta)=1$, then $\displaystyle \alpha+\beta$ is equal to