Find the next number in the series: 265, 269, 278, 284, ?

Introduction / Context:
This question presents a number series whose differences do not remain constant but still follow a structured pattern. The aim is to recognise how the increments change from term to term and then apply the same logic to find the next number after 265, 269, 278 and 284.

Given Data / Assumptions:
- Series: 265, 269, 278, 284, ?- Exactly one term, the fifth term, is missing.- The pattern is expected to involve a regular change in the size of the increments.

Concept / Approach:
When differences are not constant, candidates should compute them and see whether they themselves follow a pattern. Often the increments alternate between two sequences that grow slowly. Here, we will analyse the step sizes and check whether they form two interleaved arithmetic progressions, one for odd steps and one for even steps.

Step-by-Step Solution:
- Compute differences between consecutive terms: • 269 - 265 = 4. • 278 - 269 = 9. • 284 - 278 = 6.- Now interpret these as part of two interleaved sequences of step sizes: • Odd numbered increments: 4, 6, ... (first, third, fifth step, etc.). • Even numbered increments: 9, 11, 13, ... (second, fourth, sixth step, etc.).- The odd increments increase by 2: 4, 6, 8 and so on.- The even increments also increase by 2: 9, 11, 13 and so on.- We already have the first three increments: 4 (odd), 9 (even), 6 (odd).- The next increment should be the second even increment, which is 11.- Therefore, the missing fifth term is 284 + 11 = 295.

Verification / Alternative check:
- We can list the step sizes explicitly as: +4, +9, +6, +11, +8, +13, ... where odd positions (1, 3, 5, ...) have values 4, 6, 8, ... and even positions (2, 4, 6, ...) have values 9, 11, 13, ...- The given series 265, 269, 278, 284 aligns with +4, +9 and +6, so the next step should indeed be +11, giving 295.- No other candidate continues both interleaved arithmetic progressions so neatly.

Why Other Options Are Wrong:
- 291, 299, 303 and 305 correspond to increments of 7, 15, 19 and 21 from 284.- None of these increments fit the required values 11 or 8 that would continue the structured double progression.- Using any of these alternatives would break the regular growth in the step sizes.

Common Pitfalls:
- Some candidates look for a simple quadratic or cubic form instead of analysing the differences directly.- It is easy to miss the idea that there can be two interleaved patterns operating simultaneously.- Guessing based only on approximate growth without calculating the exact increments often leads to wrong options like 291 or 299.

Final Answer:
Continuing the structured pattern of step sizes, the next term is 284 + 11 = 295.