UPSC Prelims 2009 Q100: Science & Technology – Quantitative Aptitude
While adding the first few continuous natural numbers, a candidate missed one of the
numbers and wrote the answer as 177. What was the number missed?
- (a) 11
- (b) 12
- (c) 13
- (d) 14
Answer: (c) 13
Explanation
Finding the missed number in a natural-number sum
A candidate summed the first few continuous natural numbers but missed one, getting 177 — we need the missed number.
| Step | Reasoning |
|---|---|
| Try n=19 | Full sum = 19×20/2 = 190; missed number = 190 − 177 = 13 (valid, since 13 ≤ 19) |
| Try n=18 | Full sum = 18×19/2 = 171, which is already LESS than 177 — impossible, since removing a number can only make the total smaller, never larger |
| Conclusion | n=19 is the correct natural-number range, with the missed number being 13 |
Why n=19 is the right choice:
| Point | Detail |
|---|---|
| Full sum (1 to 19) | 190 |
| Missed number | 190 − 177 = 13 |
| Validity check | 13 is indeed within 1 to 19 |
Answer: (c) 13.
Why each option is right or wrong
- (a) Incorrect. 11 does not match the correct missed-number calculation for n=19.
- (b) Incorrect. 12 also does not match the exact calculation.
- (c) CORRECT. With n=19 (full sum 190), the missed number is 190 − 177 = 13.
- (d) Incorrect. 14 does not match the correct calculation either.
Key point
Testing n=19 gives a full sum of 190; since the candidate's total was 177, the missed number is 190 − 177 = 13 — a valid number within the 1-to-19 range.
Remember
Sum of first 19 natural numbers = 190; with 177 given, the missed number is 13.
Common mistakes
- Testing the wrong value of n, leading to a full sum that's too small or doesn't give a valid missed number.
- Forgetting to verify that the computed missed number actually falls within the valid 1-to-n range.
- Making an arithmetic error in calculating n(n+1)/2 for the trial value of n.
Exam tip
For 'missed number in a sum' puzzles, find the full sum n(n+1)/2 that's just slightly ABOVE the given (incomplete) total, then subtract to get the missed number — always verify it falls within 1 to n.
Sources
- NCERT Class XI Mathematics — Sequences and Series
- Arun Sharma — How to Prepare for Quantitative Aptitude for the CAT (Progressions)