In the sequence 125, 138, 164, ?, 255, which number should replace the question mark to maintain the pattern?

Introduction / Context:
This question involves a sequence where the increase between consecutive terms is not constant but still follows a regular pattern. Identifying a second level pattern in the differences between terms is a common technique for solving such problems in aptitude exams.

Given Data / Assumptions:
- The sequence is 125, 138, 164, ?, 255.
- Exactly one term is missing and must be found.
- The pattern is assumed to be arithmetic at the level of differences between consecutive terms.
- All numbers are positive integers.

Concept / Approach:
When differences between consecutive terms do not remain constant but still grow in a controlled way, we often have an arithmetic progression in the differences themselves. That is, if d1, d2, d3, d4 are the differences, then d2 - d1, d3 - d2, and so on may be equal. The plan is to compute the known differences and see if they follow a simple pattern such as adding a fixed number like 13 each time.

Step-by-Step Solution:
Step 1: Compute known differences. From 125 to 138: difference is 138 - 125 = 13. From 138 to 164: difference is 164 - 138 = 26. Step 2: Look for a pattern in the differences. The differences are 13 and 26, which suggests multiples of 13. We can guess the next differences as 39 and 52, increasing by 13 each time. Step 3: Use the next difference to find the missing term. Next difference after 26 is 39. So missing term = 164 + 39 = 203. Step 4: Check that the last given term 255 fits the pattern. Next difference after 39 should be 52. 203 + 52 = 255, which matches the last term.

Verification / Alternative check:
Write the full difference pattern: 125, 138, 164, 203, 255. Differences are 13, 26, 39, 52. These differences form an arithmetic progression with first term 13 and common difference 13. This gives a very neat and regular second level pattern, confirming that 203 is the only suitable value for the missing term.

Why Other Options Are Wrong:
Option A (189) would give differences 13, 26, 25, which breaks the increasing by 13 pattern.
Option B (197) produces differences that do not align with multiples of 13.
Option C (200) also fails to give a smooth arithmetic progression in the differences.
Option D (203) uniquely preserves a difference sequence of 13, 26, 39, 52, so it is the only correct choice.

Common Pitfalls:
Learners sometimes search only for a constant difference and abandon the question when none appears. Another mistake is to try random ratios instead of checking the pattern in differences. For sequences that grow moderately quickly, always compute first level differences and then check whether these differences themselves form a simple sequence like an arithmetic progression.

Final Answer:
The missing number in the sequence is 203.