In the sequence 11, 17, 39, 85, ?, determine the next number that correctly continues the pattern.

Introduction / Context:
This number series problem involves non constant differences, so it requires examining both first and second differences. The pattern is based on differences that themselves increase in a regular way, a common technique used in competitive exam series questions.

Given Data / Assumptions:

• Series: 11, 17, 39, 85, ?
• All terms are whole numbers.
• We assume there is a single hidden rule that governs the entire series.

Concept / Approach:
We start by finding the difference between successive terms. If these differences do not form a constant pattern, we then look at differences of those differences. When second differences show a simple arithmetic progression, we can extend that pattern to find the missing term in the original series.

Step-by-Step Solution:
Step 1: Compute first differences: 17 - 11 = 6, 39 - 17 = 22, 85 - 39 = 46.Step 2: The first differences are 6, 22 and 46, which again do not form a simple sequence at first glance.Step 3: Compute second differences: 22 - 6 = 16 and 46 - 22 = 24.Step 4: The second differences 16 and 24 show a pattern: they differ by 8. It suggests that the next second difference should be 32, continuing the addition of 8.Step 5: The next first difference is then 46 + 32 = 78.Step 6: Add this to the last known term: 85 + 78 = 163.

Verification / Alternative check:
Write the full structure: first differences 6, 22, 46, 78 and second differences 16, 24, 32. The second differences increase steadily by 8, which is exactly what we extended. No other option, such as 133, 143, 153 or 173, produces this neat combination of first and second difference patterns. Hence 163 is uniquely consistent.

Why Other Options Are Wrong:
Substituting 133, 143, 153 or 173 gives first and second differences that become irregular and do not follow the observed pattern of second differences increasing by 8. These choices break the smooth growth that the series exhibits, so they must be rejected.

Common Pitfalls:
Many learners stop after trying only simple operations like constant addition or multiplication and then guess. For higher level series, exam setters often rely on second differences. Failing to compute them leads to confusion and incorrect answers.

Final Answer:
The next number that correctly continues the given series is 163.