Sequences & Series
Infinite Series with Nested Geometric Fractions
nta_pyq_2023_apr
Grade 11
Question:
If the sum $\left(1-\dfrac{1}{2}\right)+\left(1-\dfrac{1}{2^2\cdot3}+\cdots\right)+\cdots=\dfrac{\alpha}{\beta}$, $\gcd(\alpha,\beta)=1$, then $\alpha+3\beta$ is equal to
Step-by-Step Solution
Key Concept: Each group is a finite alternating geometric sum. Total $=\frac{1}{a+b}\left(\frac{a^2}{1-a}-\frac{b^2}{1+b}\right)$ where $a=\frac{1}{2},b=\frac{1}{3}$.
Sum $=\frac{1}{2}$. $\alpha+3\beta=1+6=7$.
Correct Answer: 7