Sequences & Series
GP Product Sequence — α+β
nta_pyq_2026_jan
Grade 11
Question:
Let $729,81,9,1,\ldots$ be a sequence and $P_n$ denote the product of the first $n$ terms of this sequence. If $2\displaystyle\sum_{n=1}^{40}(P_n)^{1/n}=\dfrac{3^\alpha-1}{3^\beta}$ and $\gcd(\alpha,\beta)=1$, then $\alpha+\beta$ is equal to
Step-by-Step Solution
Key Concept: Terms: $3^6,3^4,3^2,3^0,\ldots$ (ratio $1/9$). $P_n=3^{n(7-n)}$. $(P_n)^{1/n}=3^{7-n}$.
$\alpha=40$, $\beta=33$. $\alpha+\beta=73$.
Correct Answer: 3