A series of letter pairs is given with one pair missing. Choose the pair that completes the pattern: ZA, XC, TG, NM, ?

Introduction / Context:
This alphabet series problem uses pairs of letters that move symmetrically through the alphabet. The first letters descend while the second letters ascend, with step sizes that gradually change. You need to find the next pair that maintains this dual pattern.

Given Data / Assumptions:

• Given pairs: ZA, XC, TG, NM, ? • English alphabet positions: A = 1 to Z = 26. • The first letters form one sequence; the second letters form another. • The step sizes between consecutive first letters and second letters may grow in a simple numeric way.

Concept / Approach:
We convert each letter to its numeric position and then examine how first letters change from Z to X to T to N, and how second letters change from A to C to G to M. When we compute the differences, a pattern emerges with step sizes −2, −4, −6 for first letters and +2, +4, +6 for second letters. We then extend these patterns to get the next pair.

Step-by-Step Solution:
Step 1: Analyze first letters: Z, X, T, N. Positions: Z = 26, X = 24, T = 20, N = 14. Differences: 26 → 24 is −2, 24 → 20 is −4, 20 → 14 is −6. The negative step size increases by 2 each time. Next difference: −8, so 14 − 8 = 6, which is F. Step 2: Analyze second letters: A, C, G, M. Positions: A = 1, C = 3, G = 7, M = 13. Differences: 1 → 3 is +2, 3 → 7 is +4, 7 → 13 is +6. The positive step size increases by 2 each time. Next difference: +8, so 13 + 8 = 21, which is U. Step 3: Combine the results to get the pair FU.

Verification / Alternative check:
The full numeric pattern is: first letters 26, 24, 20, 14, 6 with steps −2, −4, −6, −8; second letters 1, 3, 7, 13, 21 with steps +2, +4, +6, +8. This is a smooth and symmetric pattern of increasing step sizes, making FU the only consistent continuation. None of the other options fulfill both sequences at once.

Why Other Options Are Wrong:
• KL: These letters do not emerge from the −8 and +8 changes required from N and M respectively. • LM: This pair is too close to the previous pair NM and does not reflect the larger step sizes that the pattern demands. • TI: The first letter T would move backwards, and the second letter I would move forward, but both with incorrect step sizes compared to the established sequence.

Common Pitfalls:
A frequent error is to assume a constant step size for the entire sequence. Another is to guess based only on approximate positions without calculating the exact differences. When step sizes themselves evolve, careful numeric work is essential.

Final Answer:
The pair that correctly completes the sequence is FU.