Sequence and Series
Arithmetico-Geometric Progression
JEE Main 2022
Grade 11
Question:
The sum $1 + 2 \cdot 3 + 3 \cdot 3^2 + 4 \cdot 3^3 + \dots + 10 \cdot 3^9$ is equal to:
(1) $\frac{19 \cdot 3^{10} - 1}{4}$
(2) $\frac{19 \cdot 3^{10} + 1}{4}$
(3) $5 \cdot 3^{10} + 1$
(4) $2 \cdot 3^{10}$
Step-by-Step Solution
Key Concept: Let $S = \sum_{k=1}^{10} k \cdot 3^{k-1}$. Compute $3S = \sum k \cdot 3^k$, then subtract $S - 3S$ to get a G.P.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)