# Q. 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

(c) J

(d) M

Correct Answer: (b) I

#### Question from UPSC Prelims 2023 CSAT

**Explanation :**

# Middle term of the sequence

The given sequence is a pattern of letters from the English alphabet starting from Z and going backwards to A. Each letter is repeated a number of times equal to its position in the reversed alphabet (Z is 1st, Y is 2nd, X is 3rd, and so on).

The total number of terms in the sequence is calculated by adding the number of repetitions for each letter, which forms an arithmetic series: **2 (for Z) + 3 (for Y) + 4 (for X) + … + 27 (for A).**

The sum of this series is 377= (n(n+1)/2 -1), n=27

The middle term is the 189th term (since 377/2 = 188.5). To find out which letter this term is, we subtract the total number of terms for the letters before it from 189.

The total number of terms for the letters before I (starting from Z) is 188 (2 for Z, 3 for Y, 4 for X, …, 8 for S). So, 189 – 188 = 1, which means the 189th term is the first repetition of I. Therefore,

## The middle term of the sequence is I.

**Proof –** A+B+C+D+E+F+G+H=27+26+25+24+23+22+21+20=188