Mathematics Question from JEE Main 2026 · 24 January, Shift 1

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