In the alphanumeric series FK27, LQ64, RW125, which of the following terms correctly completes the pattern as the next element?

Introduction / Context:
This question presents an alphanumeric series: FK27, LQ64, RW125, followed by a missing term. Each term includes two letters and a number. Such series usually combine patterns in both the letters and the numbers. To answer correctly, we must understand how the first and second letters change and how the numerical part evolves from one term to the next.

Given Data / Assumptions:

• Given terms: FK27, LQ64, RW125.
• Letters belong to the English alphabet; positions are A 1 to Z 26.
• The numeric parts are 27, 64, and 125.
• We must choose the next term among CX216, XB216, XC216, YB343, XD216.

Concept / Approach:
We treat the letter part and the number part separately. For the number part, notice that 27, 64, and 125 are familiar perfect cubes: 3 to the power 3, 4 to the power 3, and 5 to the power 3. This strongly suggests that the next number should be the next cube, that is 6 to the power 3 equal to 216. For the letter part, we check how the first letters and second letters move in the alphabet. If both first and second letters increase by the same constant difference from term to term, we can apply that difference to find the next pair of letters.

Step-by-Step Solution:
Step 1: Examine the numbers: 27, 64, 125. These equal 3^3, 4^3, and 5^3 respectively. Step 2: Continuing this pattern, the next integer after 5 is 6, and 6^3 equals 216. So the next numeric term should be 216. Step 3: Look at the first letters of each term: F, L, R. Their positions are F 6, L 12, and R 18. Step 4: From 6 to 12 is an increase of +6, and from 12 to 18 is another +6. So the first letters increase by 6 positions each time. Step 5: Add 6 to 18 to get 24, which corresponds to the letter X. Hence the next first letter is X. Step 6: Now consider the second letters: K, Q, W. K is 11, Q is 17, W is 23. Step 7: From 11 to 17 is +6, and from 17 to 23 is +6. So the second letters also increase by 6 positions each time. Step 8: Add 6 to 23 to get 29. In a 26 letter alphabet, going 29 means 26 plus 3, so we wrap around and get the 3rd letter C. Step 9: Therefore, the next letter pair should be X and C, forming "XC". Step 10: Combine the letter pair "XC" with the numeric pattern "216" to obtain the term "XC216".

Verification / Alternative check:
We can verify by writing the entire extended series: FK27 (6,11,3^3), LQ64 (12,17,4^3), RW125 (18,23,5^3), XC216 (24,3,6^3). The first letter jumps by +6 every time. The second letter also jumps by +6 in a modular way around the alphabet. The numbers follow cubes of consecutive integers. Additionally, none of the other options matches the combination of XC and 216. Thus "XC216" is the only term that maintains both the letter and number patterns.

Why Other Options Are Wrong:
"CX216" has the letters reversed and breaks the sequence of first and second letters based on position increments.
"XB216" keeps the first letter X but uses B instead of C, which does not correspond to the correct wrap around after adding 6 to W (23).
"YB343" contains 343 (7^3) instead of 216, which would skip one step in the cube pattern and also start the numbers too late.
"XD216" has the correct numeric part but the second letter D is not consistent with the previous increments of plus six.

Common Pitfalls:
Candidates sometimes focus only on the numeric part and ignore the letter pattern, or vice versa. Another trap is to miscalculate modular arithmetic when the alphabet index goes beyond 26. It is important to treat each component with equal attention, check patterns thoroughly, and remember wrapping behaviour for letters after Z, if needed.

Final Answer:
The correct term that completes the series is XC216.