In the following number series, one term is incorrect. Identify the wrong term: 3, 10, 27, 4, 16, 64, 5, 25, 125.

Introduction / Context:
This problem requires identifying the term that breaks an underlying pattern in a structured series. The series is clearly grouped in sets of three numbers, and within each group there is a strong hint of powers, particularly cubes. Recognizing the pattern within each group is the key to discovering the incorrect term.

Given Data / Assumptions:
The series is:
3, 10, 27, 4, 16, 64, 5, 25, 125
We can naturally divide these into groups: (3, 10, 27), (4, 16, 64), and (5, 25, 125). We assume that each group should follow the same rule relating the three numbers, and we look for any group that does not fit that rule.

Concept / Approach:
The pattern suggests that each group may be based on a number and its powers. For example, 3 and 27 relate as 3^1 and 3^3, while 4 and 64 relate as 4^2 and 4^3, and similarly 5 and 125 correspond to 5^2 and 5^3. The middle number in each triplet should therefore match the square of the first element in that group. We test this idea for all three groups.

Step-by-Step Solution:
Step 1: Examine the second group: (4, 16, 64).Here, 4 is the base, 16 is equal to 4^2, and 64 is equal to 4^3. This group is perfectly consistent.Step 2: Examine the third group: (5, 25, 125).Here, 5 is the base, 25 is 5^2, and 125 is 5^3. This group is also correct.Step 3: Examine the first group: (3, 10, 27).Here, 3 is the base and 27 is 3^3, but 10 is not equal to 3^2. The correct square would be 9, not 10.Step 4: Therefore, 10 is the only term that fails to match the consistent rule observed in the other groups.

Verification / Alternative check:
Rewriting the series with the intended pattern gives us (3, 9, 27), (4, 16, 64), and (5, 25, 125), which shows a very clean structure: n, n^2, n^3 for n equal to 3, 4, and 5. Because only the middle number of the first triplet is wrong, the term 10 must be the one that does not belong in the original series.

Why Other Options Are Wrong:
The numbers 3, 4, 16, 27, 25, 64, and 125 all fit correctly into the n, n^2, n^3 pattern for n equal to 3, 4, or 5. Removing any of these would destroy the otherwise perfect structure. Only 10 fails to correspond to the correct square of 3 and thus stands out as incorrect.

Common Pitfalls:
Some students may look at the series as a single linear progression rather than as triplets based on powers. They might then focus on differences instead of the much clearer power relationship and become confused. Always consider grouping terms when a pattern appears to repeat every few numbers.

Final Answer:
The wrong term in the given series is 10.