Question

Determine the 288th term of the series a,b,b,c,c,c,d,d,d,d,e,e,e,e,e,f,f,f,f,f,f... :

• A. X
• B. Z
• C. V
• D. M
• E. U

Hint:

Use triangular numbers to find which letter block contains the desired position.

Answer 1

The Correct Option is A

Step 1: Understanding the Pattern:
The series: a (1 time), b (2 times), c (3 times), d (4 times), e (5 times), f (6 times), g (7 times), and so on.
The number of terms up to the letter that appears \(n\) times is: \[ S_n = \frac{n(n+1)}{2} \]

Step 2: Finding the Letter:
We need the 288th term. Find \(n\) such that: \[ S_{n-1} < 288 \leq S_n \]
\[ S_{23} = \frac{23 \times 24}{2} = 276 \]
\[ S_{24} = \frac{24 \times 25}{2} = 300 \]
Since \(276 < 288 \leq 300\), the 288th term belongs to the block of the 24th letter.
Alphabet: a = 1, b = 2, c = 3, ..., 24th letter = X.

Final Answer: X