A series of two letter groups is given as FU, LO, RI, ? Using positions in the English alphabet, which pair of letters should come next to continue this pattern?

Introduction / Context:
This question presents a two letter series: FU, LO, RI, followed by a missing pair. Problems like this test the ability to track patterns in both letters of the pair simultaneously. The trick here is that the first letters move forward in the alphabet, while the second letters move backward, each with a constant step size.

Given Data / Assumptions:

• Series: FU, LO, RI, ? • Options: PK, XC, UF, RI • Alphabet positions: F 6, L 12, R 18, X 24; U 21, O 15, I 9, C 3. • Assumption: There is a consistent numeric pattern governing both the first and the second letters separately.

Concept / Approach:
We convert each letter to its position in the alphabet and consider the first letters F, L, R as one sequence and the second letters U, O, I as another. Then we calculate differences between successive terms. If we find that the differences are constant, we can extend each sequence by applying the same step. This gives the positions of the letters in the missing pair, which we convert back to letters and match to the available options.

Step-by-Step Solution:
Step 1: First letters: F, L, R. Their positions are 6, 12, and 18. Step 2: Differences between positions are 12 − 6 = 6 and 18 − 12 = 6. So the first letter series increases by 6 each time. Step 3: Extend forward: 18 + 6 = 24. Position 24 is X, so the next first letter should be X. Step 4: Second letters: U, O, I. Their positions are 21, 15, and 9. Step 5: Differences are 15 − 21 = −6 and 9 − 15 = −6. So the second letter series decreases by 6 each time. Step 6: Extend backward: 9 − 6 = 3. Position 3 is C, so the next second letter should be C. Step 7: Combining results, the missing pair is XC.

Verification / Alternative check:
We can verify visually by writing positions as pairs: FU is (6,21), LO is (12,15), RI is (18,9), and the next term must be (24,3). We see that the first coordinate increases by 6 and the second decreases by 6 at each step. Converting (24,3) back to letters gives X and C, that is XC. Checking each option confirms that only XC has letters at positions 24 and 3 in that order.

Why Other Options Are Wrong:
PK corresponds to positions (16,11), which do not fit a +6 first coordinate and −6 second coordinate rule starting from (18,9). UF repeats the original first term rotated, breaking the consistent numeric progression. RI is already given in the series as the third term and cannot be the next term.

Common Pitfalls:
A common mistake is to look for a direct relation between the two letters of each pair instead of treating them as two independent sequences. Another error is to miscount the numeric positions of letters near the end of the alphabet, especially X. Carefully writing the alphabet with numbers and noting plus six and minus six changes prevents these calculation errors.

Final Answer:
The pair of letters that correctly continues the series FU, LO, RI is XC.