A telescoping sum — Number System Aptitude Question
A telescoping sum: Find the value of: 1/(1×2) + 1/(2×3) + 1/(3×4) + … + 1/(9×10) Answer and worked solution on CodeKairo.
- Topic: Number System
- Section: Quantitative Aptitude
- Difficulty: Medium
- Time target: 75 seconds
Question
Find the value of:
1/(1×2) + 1/(2×3) + 1/(3×4) + … + 1/(9×10)
Options
- 1/10
- 1/2
- 9/10
- 1
Show the answer and the worked solution
Correct answer: C. 9/10
Worked solution
- 1/(n(n + 1)) = 1/n − 1/(n + 1).
- The sum becomes (1/1 − 1/2) + (1/2 − 1/3) + … + (1/9 − 1/10).
- Everything cancels except 1 − 1/10 = 9/10.
Answer: 9/10.
The idea
Splitting a product denominator into a difference makes the middle terms cancel in pairs. The general result is 1 − 1/(n + 1) = n/(n + 1) for the sum up to 1/(n(n+1)).
Shortcut. The first term alone is 1/2 and the endless series sums to 1, so the answer lies strictly between 1/2 and 1 — only 9/10 does.
Common trap. 1/10 — keeping the leftover 1/10 from the cancellation and forgetting the leading 1, so 1 − 1/10 becomes just 1/10.