A series of letter pairs is given with one term missing. Choose the correct alternative that will complete the series: AZ, CX, FU, ?

Introduction / Context:
This question belongs to the topic of letter series. Instead of single letters, here we have ordered pairs of letters: AZ, CX, FU and a missing pair. The task is to identify the pattern governing changes in the first and second letters and then extend that pattern to find the next pair. Such questions test pattern recognition, familiarity with alphabetical positions and the ability to handle simultaneous progressions.

Given Data / Assumptions:
- Given pairs: AZ, CX, FU, ?.
- All letters are English capital letters from A to Z.
- We assume standard alphabetical positions: A as 1, B as 2, ..., Z as 26.
- The pattern may involve arithmetic progressions on positions of first and second letters.

Concept / Approach:
The key approach is to separate the pair into two independent sequences: one made from the first letters (A, C, F, ?) and one from the second letters (Z, X, U, ?). We then compute differences between consecutive letters by their positions in the alphabet. If we see a simple increasing or decreasing pattern, we extend it to find the next term. Note that the first sequence may be increasing while the second decreases, which is common in such series.

Step-by-Step Solution:
Step 1: Consider the first letters: A, C, F. Their positions are A = 1, C = 3, F = 6. Step 2: Calculate the differences: from 1 to 3 is +2, from 3 to 6 is +3. Step 3: The increase pattern in the first letters is +2, then +3. A natural continuation is to increase by +4 next. Step 4: Add 4 to the position of F (which is 6). So 6 + 4 = 10. The 10th letter is J. So the next first letter is J. Step 5: Now examine the second letters: Z, X, U. Positions are Z = 26, X = 24, U = 21. Step 6: Differences are 26 to 24 is -2, and 24 to 21 is -3. So the second letters follow a decreasing pattern: -2, then -3. Step 7: To continue the pattern, subtract 4 next. From 21 subtract 4 to get 17. The 17th letter is Q. Step 8: Therefore the next pair in the series is JQ.

Verification / Alternative check:
An alternative way to see the pattern is to write the pairs as (1,26), (3,24), (6,21) in terms of positions. The first coordinate increases by +2, then +3, then +4. The second coordinate decreases by -2, then -3, then -4. This symmetric pattern (+2, +3, +4 and -2, -3, -4) confirms that JQ, corresponding to (10,17), is the correct continuation. Substituting JQ back into the list AZ, CX, FU, JQ shows a smooth and consistent pattern in alphabetical distances.

Why Other Options Are Wrong:
Option A (IR) would give first letter I (9) which is +3 from F, not +4, and second letter R (18) which does not fit the -4 progression. Option B (IV) breaks both the first and second letter patterns. Option D (KP) uses K (11) instead of J and P (16), which do not match the expected positions 10 and 17. Option E (HW) also mismatches the expected pattern. Only option C (JQ) maintains the exact arithmetic progression for both letters.

Common Pitfalls:
A common mistake is to look for a simple constant difference in the letters and give up when the difference is not constant. Here the pattern is a growing step size, which some candidates overlook. Another pitfall is to treat the pair as a single unit and search for obscure patterns instead of separating it into two easier sequences. Always examine each position of a letter pair or triplet separately, and check whether the differences form an arithmetic or other recognizable progression.

Final Answer:
The missing pair that completes the series AZ, CX, FU, ? is JQ.