In the series 114, 131, 165, 216, ?, 369, what number should replace the question mark?

Introduction / Context:
This question involves a number series where the jumps between terms are not constant but follow a secondary pattern. The focus is on recognising an arithmetic progression in the differences between consecutive terms.

Given Data / Assumptions:

• Series: 114, 131, 165, 216, ?, 369.
• Exactly one term between 216 and 369 is missing.
• The same rule applies to all steps in the series.

Concept / Approach:
We start by computing differences between known adjacent terms. If those differences themselves grow in a regular way, such as forming an arithmetic progression, that pattern can be extended to locate the missing term.

Step-by-Step Solution:
Step 1: Compute differences for the known parts: 131 − 114 = 17. Step 2: Next difference: 165 − 131 = 34. Step 3: Next difference: 216 − 165 = 51. Step 4: Observe that 17, 34, and 51 form a pattern where each difference increases by 17: 34 − 17 = 17 and 51 − 34 = 17. Step 5: Following this rule, the next difference should be 51 + 17 = 68. Step 6: Add this to the last known term: missing term = 216 + 68 = 284. Step 7: To confirm, check the final difference: 369 − 284 = 85, which is again 68 + 17, continuing the same pattern.

Verification / Alternative check:
With the missing term set to 284, the series becomes 114, 131, 165, 216, 284, 369. The differences are 17, 34, 51, 68, 85. This is an arithmetic progression of differences where each difference increases by 17. The rule fits every step, confirming that 284 is the correct value.

Why Other Options Are Wrong:
If we insert 294, 304, or 314, the last two differences do not form a difference sequence that increases by a constant 17. For example, with 294, the differences would be 17, 34, 51, 78, 75, which destroys the linear pattern. Hence those options cannot be correct.

Common Pitfalls:
One pitfall is to assume that the main series itself is an arithmetic progression and look for a constant difference, which clearly does not work here. Another mistake is to stop after noticing that the differences roughly increase and not to compute the exact increments, which are crucial in this question. Always compute all differences carefully to find the precise pattern.

Final Answer:
The number that correctly completes the series is 284.