What is the middle term of the sequence Z, Z, Y, Y, Y, X, X, X, X, W, W, W, W, W, ..., A ?
- (a) H
- (b) I ✓ UPSC's answer
- (c) J
- (d) M
Why the answer is (b)
• Z appears 2 times, Y 3 times, X 4 times and so on, so the k-th letter from Z appears k + 1 times, down to A (the 26th) appearing 27 times.
• Total terms = 2 + 3 + … + 27 = 377, so the middle term is the 189th.
• The first k letters together fill k(k + 3)/2 places: 17 letters (Z to J) fill 170 places and 18 letters (Z to I) fill 189.
• So the 189th term is the last I — option (b).
Why the other options are wrong
- (a) H
- The H block begins only after the 189th term (terms 190 to 209).
- (c) J
- The J block ends at the 170th term, before the middle.
- (d) M
- The M block ends at the 104th term, well before the middle.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2023, held on 28 May 2023. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.