In the series AABABCABCDABCDE..., which letter appears at the 100th place?
- (a) G
- (b) H
- (c) I ✓ UPSC's answer
- (d) J
Why the answer is (c)
• The series AABABCABCDABCDE... is built by concatenating groups where the n-th group contains the first n letters of the alphabet: Group 1 = A, Group 2 = AB, Group 3 = ABC, Group 4 = ABCD, and so on.
• The cumulative number of letters after completing n groups is 1 + 2 + 3 + … + n = n(n+1)/2.
• Setting n(n+1)/2 ≥ 100: for n = 13, 13×14/2 = 91; for n = 14, 14×15/2 = 105. Hence the 100th position falls inside Group 14, which spans positions 92 through 105.
• The offset of the 100th letter within Group 14 is 100 − 91 = 9, meaning it is the 9th letter of that group.
• Group 14 reads A, B, C, D, E, F, G, H, I, J, K, L, M, N; its 9th letter is I.
• Therefore the letter at the 100th place is I, corresponding to option (c).
Why the other options are wrong
- (a) G
- G is the 7th letter of the alphabet, which would place it at position 91 + 7 = 98 in the series, not 100.
- (b) H
- H is the 8th letter of the alphabet, which would place it at position 91 + 8 = 99 in the series, not 100.
- (d) J
- J is the 10th letter of the alphabet, which would place it at position 91 + 10 = 101 in the series, not 100.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2022, held on 5 June 2022. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.