Let \(729,81,9,1, \ldots\) be a sequence and \(P_n\) denote the product of the first \(n\) terms of this sequence. If \(…

Let \(729,81,9,1, \ldots\) be a sequence and \(P_n\) denote the product of the first \(n\) terms of this sequence.

If \(2\sum _{n=1}^{40}{\left({P}_{n}\right)}^{\frac{1}{n}}=\frac{{3}^{\alpha }−1}{{3}^{\beta }}\) and \(\operatorname{gcd}(\alpha, \beta)=1\), then \(\alpha+\beta\) is equal to

[JEE Main 2026, 24 Jan (Shift 1)]