Letter Series Pattern Breakdown

The question asks us to find the missing four-letter cluster in the given sequence: CADI, EAFI, GAHI, ____, KALI. We need to identify the underlying pattern to determine the correct cluster for the blank space.

Analyzing Positional Changes

To solve this, we can examine the progression of letters at each specific position within the four-letter clusters:

• First Letter: C → E → G → ? → K
• Second Letter: A → A → A → ? → A
• Third Letter: D → F → H → ? → L
• Fourth Letter: I → I → I → ? → I

Step-by-Step Pattern Identification

Let's analyze the pattern for each position individually:

• First Letter Progression:

  The sequence of first letters is C, E, G, ?, K. To understand the pattern, let's consider their numerical positions in the English alphabet:

  • C is the 3^rd letter.
  • E is the 5^th letter.
  • G is the 7^th letter.
  • K is the 11^th letter.

  We can see a consistent increase of 2 positions between each letter: $3 + 2 = 5$ (C to E), $5 + 2 = 7$ (E to G). Following this arithmetic progression, the next letter should be at position $7 + 2 = 9$. The 9^th letter of the alphabet is I. To confirm, the position after that should be $9 + 2 = 11$, which corresponds to 'K', matching the last term in the series. Thus, the first letter of the missing cluster is I.

• Second Letter Consistency:

  The second letters in the series are A, A, A, ?, A. The letter 'A' remains unchanged throughout the sequence. Therefore, the second letter of the missing cluster is also A.

• Third Letter Progression:

  The sequence of third letters is D, F, H, ?, L. Let's look at their alphabetical positions:

  • D is the 4^th letter.
  • F is the 6^th letter.
  • H is the 8^th letter.
  • L is the 12^th letter.

  Similar to the first letter, there is an increment of 2 positions: $4 + 2 = 6$ (D to F), $6 + 2 = 8$ (F to H). Applying this pattern, the next letter should be at position $8 + 2 = 10$. The 10^th letter is J. The subsequent position is $10 + 2 = 12$, which correctly corresponds to 'L', the final term's third letter. So, the third letter of the missing cluster is J.

• Fourth Letter Consistency:

  The fourth letters are I, I, I, ?, I. Like the second letter, the letter 'I' is constant in this position. Therefore, the fourth letter of the missing cluster is I.

Determining the Missing Cluster

By assembling the letters determined for each position, we find the missing cluster:

• First letter: I
• Second letter: A
• Third letter: J
• Fourth letter: I

Combining these gives us the missing letter cluster: IAJI.