UPSC Prelims 2008 Q1: Science & Technology – Logical Reasoning & Puzzles
In the series AABABCABCDABCDE... which letter occupies the 100th position?
- (a) H
- (b) I
- (c) J
- (d) K
Answer: (b) I
Explanation
Finding the 100th letter in a growing-block series
The series AABABCABCDABCDE... is built from successive blocks: A, AB, ABC, ABCD, ABCDE, ..., where block *n* has *n* letters (A through the *n*th letter).
| Step | Calculation |
|---|---|
| Cumulative length after block n | n(n+1)/2 |
| After block 13 | 13×14/2 = 91 |
| After block 14 | 14×15/2 = 105 |
| Position 100 falls in | Block 14 (positions 92 to 105), since 91 < 100 ≤ 105 |
| Position within block 14 | 100 − 91 = 9th letter of block 14 |
| Block 14's letters | A, B, C, D, E, F, G, H, I, J, K, L, M, N — the 9th is I |
Why the triangular-number formula applies:
| Point | Detail |
|---|---|
| Block n length | n letters (A to the nth letter of the alphabet) |
| Cumulative total | n(n+1)/2, the triangular number formula |
Answer: (b) I.
Why each option is right or wrong
- (a) Incorrect. H would be the 8th letter of block 14, one position too early.
- (b) CORRECT. Position 100 falls at the 9th letter of block 14, which is I.
- (c) Incorrect. J would be the 10th letter of block 14, one position too late.
- (d) Incorrect. K overshoots the correct position within block 14.
Key point
The series is built from blocks of growing length (A, AB, ABC, ...); using the triangular-number formula n(n+1)/2 to locate block 14, position 100 lands on the 9th letter of that block, which is I.
Remember
Series AABABCABCD...: the 100th position falls on the 9th letter of block 14, which is I.
Common mistakes
- Miscounting which block position 100 falls into, due to an arithmetic slip in the triangular-number formula.
- Forgetting that each block restarts from A, not continuing the alphabet from where the previous block left off.
- Off-by-one errors when counting the position within the identified block.
Exam tip
For growing-block letter series, use the triangular-number formula n(n+1)/2 to find which block a given position falls into, then count within that block.
Sources
- NCERT Class XI Mathematics — Sequences and Series
- R. S. Aggarwal — A Modern Approach to Verbal and Non-Verbal Reasoning (Series Completion)