Sequences & Series
Telescoping Sum — 50d Value
nta_pyq_2024_apr
Grade 11
Question:
If the sum of the series $\dfrac{1}{1\cdot(1+d)}+\dfrac{1}{(1+d)(1+2d)}+\cdots+\dfrac{1}{(1+9d)(1+10d)}$ is equal to 5, then 50d is equal to:
Step-by-Step Solution
Key Concept: Each term telescopes: $\frac{1}{(1+kd)(1+(k+1)d)}=\frac{1}{d}\left(\frac{1}{1+kd}-\frac{1}{1+(k+1)d}\right)$. Sum $=\frac{1}{d}\left(1-\frac{1}{1+10d}\right)=\frac{10}{1+10d}=5$.
$d=1/10$. $50d=5$.
Correct Answer: 2