Q65. What is the middle terms of the sequence Z, Z, Y, Y, Y, X, X, X, X, W, W, W, W, W,…….,A?
Detailed Solution
The sequence consists of letters of the alphabet, where each letter appears a number of times equal to (28 - its position in the alphabet from A). So, Z (26th letter) appears 2 times, Y (25th) appears 3 times, ..., A (1st) appears 27 times.
The total number of terms in the sequence is the sum 2 + 3 + 4 + ... + 27. This is an arithmetic series sum: (number of terms / 2) * (first term + last term) = (26/2) * (2 + 27) = 13 * 29 = 377.
The middle term is (377 + 1) / 2 = 189th term. To find the 189th term, we sum the occurrences from Z downwards: Z (2), Y (3), X (4), W (5), V (6), U (7), T (8), S (9), R (10), Q (11), P (12), O (13), N (14), M (15), L (16), K (17), J (18), I (19).
Summing these: 2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18 = 170. The next letter is 'I', which appears 19 times. The 189th term falls within the 'I' block (171st to 189th term). Thus, the middle term is I.
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