Step 1: Understanding the Concept:
The series consists of 'a' repeated 1 time, 'b' repeated 2 times, 'c' repeated 3 times, and so on. The total number of terms up to the end of the \(n\)-th letter group is the sum of the first \(n\) natural numbers. Step 2: Key Formula or Approach:
Total terms up to letter \(n\) = \( \frac{n(n+1)}{2} \).
We need to find \(n\) such that \( \frac{n(n+1)}{2} \ge 288 \). Step 3: Detailed Explanation:
1. Solve for \(n\):
\( n^2 + n \ge 576 \)
Estimate \(\sqrt{576} = 24\).
2. Check \(n = 23\):
Sum = \( \frac{23 \times 24}{2} = 23 \times 12 = 276 \).
So the 276th term is the last 'w' (23rd letter).
3. The next group (24th letter) is 'x'. It starts at term 277 and goes up to term \( \frac{24 \times 25}{2} = 300 \).
4. Since 288 falls between 277 and 300, the 288th term is 'x'. Step 4: Final Answer:
The 288th term is x.