Letter Pattern Analysis

We analyze the pattern by looking at the alphabetical positions of the letters in each cluster:

• KFU to IIQ: The transformations are K (11) $\rightarrow$ I (9) [$-2$], F (6) $\rightarrow$ I (9) [$+3$], U (21) $\rightarrow$ Q (17) [$-4$].
• IIQ to GLM: The transformations are I (9) $\rightarrow$ G (7) [$-2$], I (9) $\rightarrow$ L (12) [$+3$], Q (17) $\rightarrow$ M (13) [$-4$].
• GLM to EOI: The transformations are G (7) $\rightarrow$ E (5) [$-2$], L (12) $\rightarrow$ O (15) [$+3$], M (13) $\rightarrow$ I (9) [$-4$].

The pattern for the letters is consistent: subtract 2 from the first letter's position, add 3 to the second, and subtract 4 from the third.

Applying this pattern to the last known cluster, EOI:

• First letter: E (position 5). Calculate $5 - 2 = 3$. The 3rd letter is C.
• Second letter: O (position 15). Calculate $15 + 3 = 18$. The 18th letter is R.
• Third letter: I (position 9). Calculate $9 - 4 = 5$. The 5th letter is E.

The resulting letter cluster is CRE.

Number Pattern Analysis

We examine the sequence of numbers:

• 15
• 22
• 31
• 42

Calculate the differences between consecutive numbers:

• $22 - 15 = 7$
• $31 - 22 = 9$
• $42 - 31 = 11$

The differences are increasing odd numbers (7, 9, 11). The next difference should be $11 + 2 = 13$.

Calculate the next number in the series:

• $42 + 13 = 55$

The resulting number is 55.

Conclusion

Combining the derived letter cluster (CRE) and the calculated number (55), the missing element in the series is CRE-55.

This matches Option A.