A three letter alphabet series is given with one term missing. Select the correct alternative from the given options that will complete the series HPA, FMZ, DJY, BGX, ?

Introduction / Context:
This question is a three letter alphabet series problem where each term is a group of three letters. The goal is to identify the transformation pattern from one term to the next in order to predict the missing three letter group. Such problems are slightly more complex than basic two letter series because they may involve different rules for each position within the group.

Given Data / Assumptions:

• The given sequence is HPA, FMZ, DJY, BGX, ?
• Each term consists of three capital letters from the English alphabet.
• The next term must be obtained by following a consistent pattern for each letter position.
• Alphabet positions are taken as A = 1 to Z = 26 for calculation purposes.

Concept / Approach:
A standard approach is to split each three letter term into first, second, and third letters and analyse each position as a separate sequence. If we can identify a regular arithmetic step, such as adding or subtracting a fixed value modulo 26, then we extend that pattern to determine the letters in the missing term. We must ensure that the same rule is applied consistently through the sequence.

Step-by-Step Solution:
Step 1: List the first letters: H, F, D, B. Their positions are H = 8, F = 6, D = 4, B = 2. Step 2: Observe that each first letter decreases by 2: 8 to 6, 6 to 4, and 4 to 2. So the next first letter is 2 minus 2 which is 0, and wrapping around the alphabet means we go to position 26, which is Z. Step 3: Now list the second letters: P, M, J, G. Their positions are P = 16, M = 13, J = 10, G = 7. The difference each time is minus 3. Step 4: Apply the same step again: 7 minus 3 gives 4, which corresponds to the letter D. So the second letter of the missing term is D. Step 5: Now consider the third letters: A, Z, Y, X. Their positions are A = 1, Z = 26, Y = 25, X = 24. Step 6: From A to Z is effectively a shift of minus 1 if we move backward and wrap, and then each subsequent step is also minus 1: 26 to 25, then 25 to 24. So the next step is 24 minus 1 which is 23, corresponding to W. Step 7: Combining the three positions, we get Z (first), D (second), W (third), so the missing term is ZDW.

Verification / Alternative check:
We can verify quickly by recomputing the differences: first letters always minus 2, second letters always minus 3, and third letters always minus 1 treating the alphabet cyclically. All given terms and the derived term ZDW respect these steps. None of the other options simultaneously satisfy all three positional patterns.

Why Other Options Are Wrong:
ZEV has the correct first letter Z but the second and third letters do not follow the minus 3 and minus 1 patterns. YDW and YEV fail already in the first letter because the first letter must be Z by the minus 2 sequence. The option none of these is incorrect because ZDW perfectly matches the observed rule.

Common Pitfalls:
A common error is to look only at the first letter sequence and ignore the second and third letters, leading to incomplete or incorrect reasoning. Some students also forget to wrap around from A to Z or handle it incorrectly. Another mistake is assuming the same difference for all three letter positions, which does not hold in this question. Careful separate analysis of each position avoids these mistakes.

Final Answer:
The three letter group that correctly completes the series is ZDW.