a, b, b, c, c, c, d, d, d, d, . . . . . . Find the 288th letter of this series.

Concept

In this letter series, each letter of the alphabet is repeated as many times as its position in the alphabet — a appears once, b twice, c three times, and so on. The cumulative count of letters through the nth letter of the alphabet is therefore the sum of the first n natural numbers, given by n(n+1)/2. To find which letter occupies a given position, find the smallest n for which n(n+1)/2 is greater than or equal to that position.

Application

1. Identify the pattern: the nth letter of the alphabet repeats n times, so the cumulative count of letters through the nth letter is n(n+1)/2.

2. Compute for n = 23: 23×24/2 = 276. So the first 276 letters cover the complete blocks of the 1st through 23rd letters (a through w).

3. Compute for n = 24: 24×25/2 = 300. So positions 277 through 300 all belong to the 24th letter's repeated block.

4. Since 276 < 288 ≤ 300, position 288 falls inside the 24th letter's block.

5. The 24th letter of the alphabet is x.

Cross-check

288 − 276 = 12, and 12 is within the block size of 24 for n = 24, confirming position 288 indeed lies inside the 24th letter's repeated block — the 288th letter is x.