Sequences & Series
GP Summation with Powers of 2 and 3
nta_pyq_2026_jan
Grade 11
Question:
$\dfrac{6}{3^{26}}+\dfrac{10\cdot1}{3^{25}}+\dfrac{10\cdot2}{3^{24}}+\dfrac{10\cdot2^2}{3^{23}}+\cdots+\dfrac{10\cdot2^{24}}{3}$ is equal to:
$2^{26}$
$3^{25}$
$3^{26}$
$2^{25}$
Step-by-Step Solution
Key Concept: Rewrite sum as $\tfrac{6}{3^{26}}+\tfrac{10}{3^{25}}\sum_{k=0}^{24}\left(\tfrac{2}{3}\right)^k\cdot3^{0}$... Convert: $\tfrac{10\cdot2^k}{3^{25-k}}=\tfrac{10\cdot6^k}{3^{25}}$.
$2^{26}$.
Correct Answer: 1