In the following letter analogy, FGH is related to a certain group of letters in the same way that QRS is related to SUW. Using this pattern, which group of letters should complete the analogy FGH : ? :: QRS : SUW?

Introduction / Context:
This analogy problem involves detecting a pattern of increasing shifts in consecutive letters. The relation between QRS and SUW follows a systematic set of forward jumps in the alphabet. Once we understand that pattern, we apply the same sequence of jumps to FGH to find the missing group of letters. This type of problem tests accurate counting and pattern replication in the alphabet series.

Given Data / Assumptions:
We know QRS is related to SUW. We must find a group related to FGH so that FGH : ? :: QRS : SUW. All letters are capital English letters from A to Z. The same step pattern is applied position wise in both pairs.

Concept / Approach:
We first convert the letters in QRS and SUW to their positions in the alphabet, then study the differences between corresponding letters. If we notice a consistent pattern like +2, +3, +4 for the first, second, and third letters, we can adopt that as the transformation rule. We then apply these exact differences to FGH in order to obtain the new group. Consistency across all positions is crucial.

Step-by-Step Solution:
Step 1: Determine positions for QRS and SUW. Q = 17, R = 18, S = 19. S = 19, U = 21, W = 23. Step 2: Compute the shifts for each position. First letter: 17 to 19 is +2. Second letter: 18 to 21 is +3. Third letter: 19 to 23 is +4. Thus the pattern is +2, then +3, then +4 for the three letters in order. Step 3: Apply the same +2, +3, +4 pattern to FGH. F = 6, plus 2 gives 8, which is H. G = 7, plus 3 gives 10, which is J. H = 8, plus 4 gives 12, which is L. Therefore, FGH transforms into HJL. Step 4: Match HJL with the options. HJL appears as one of the options, so it is the correct answer.

Verification / Alternative check:
We can verify the correctness of our pattern by applying it in reverse. If we subtract 2, 3, and 4 from the letters of SUW, we should get back QRS. S minus 2 is Q, U minus 3 is R, and W minus 4 is S, which matches perfectly. Likewise, subtracting 2, 3, and 4 from HJL yields FGH, confirming that HJL is the correct corresponding group. No other option would give this exact mirrored behavior.

Why Other Options Are Wrong:
OPQ, GHI, HIJ, and IKM do not result from applying the +2, +3, +4 shifts to FGH and therefore do not match the pattern derived from QRS to SUW. Some of these options represent simple consecutive sequences that ignore the increasing step structure of the analogy.

Common Pitfalls:
One common mistake is to assume that the pattern is always +1, +1, +1 or some other constant shift. Another pitfall is to look only at the first and last letters and ignore the middle one, which often carries crucial information about the step pattern. In multi letter analogies, always compare each corresponding position and confirm that the differences are consistent before applying the pattern to the new group.

Final Answer:
The group of letters that correctly completes the analogy is HJL.