JEE Mains · Maths · STD 11 - 8. sequence and series
Let 729, 81, 9, 1, .... be a sequence and \( P_{n} \) denote the product of the first n terms of this sequence. If \( 2\sum_{n=1}^{40}(P_{n})^{\frac{1}{n}}=\frac{3^{\alpha}-1}{3^{\beta}} \) and \( \gcd(\alpha,\beta)=1 \), then \( \alpha+\beta \) is equal to
- A 73
- B 74
- C 75
- D 76
Answer & Solution
Correct Answer
(A) 73
Step-by-step Solution
Detailed explanation
\(P_n=729\cdot81\cdot9 \ldots \ldots .(n\) terms \()\) \(\begin{array}{l}=3^6 \cdot 3^4 \cdot 3^2 \ldots \ldots \cdot 3^{-2 n+8} \\ P_n=3^{6+4+2+\ldots \ldots+(-2 n+8)}=3^{n(7-n)} \\ P_n^{1 / n}=3^{7-n}\end{array}\)…
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