A number series is given with one term missing. Choose the correct alternative from the options that will complete the series: 13, 25, 49, 85, ?

Introduction / Context:
This question presents an increasing number series where the differences between terms gradually grow in a structured way. By examining those differences, we can identify a pattern of increasing increments and use it to determine the missing term. This type of series evaluates the ability to track and extend non-constant differences.

Given Data / Assumptions:
Given series: 13, 25, 49, 85, ?The fifth term is missing.We assume a simple pattern in the differences between consecutive terms, likely involving multiples of a basic value.

Concept / Approach:
The method is to calculate the differences between consecutive terms and observe how these differences themselves change. If the differences increase by a constant amount, then the second differences are constant, indicating a quadratic-like pattern. Once we identify that the differences form a sequence such as 12, 24, 36, and so on, we can predict the next difference and hence the missing term.

Step-by-Step Solution:
Step 1: Compute 25 - 13 = 12.Step 2: Compute 49 - 25 = 24.Step 3: Compute 85 - 49 = 36.Step 4: The differences are 12, 24, 36, which are multiples of 12: 1 * 12, 2 * 12, 3 * 12.Step 5: The next difference should naturally be 4 * 12 = 48.Step 6: Add this difference to the last known term: 85 + 48 = 133.Step 7: Therefore, the missing term is 133.

Verification / Alternative check:
Insert the candidate term: 13, 25, 49, 85, 133. Recalculate the differences: 25 - 13 = 12, 49 - 25 = 24, 85 - 49 = 36, and 133 - 85 = 48. These differences form a clear sequence of 12, 24, 36, 48, which increases by 12 each time. This confirms that the series follows a pattern of adding successive multiples of 12, and 133 fits perfectly as the next term.

Why Other Options Are Wrong:
Option 331 would create a jump of 246 from 85, which does not match the controlled increase seen in the earlier differences.Option 132 produces a difference of 47, which breaks the multiple-of-12 pattern and the constant increase of 12 in the differences.Option 381 results in a very large and unstructured jump that is completely inconsistent with the earlier pattern.

Common Pitfalls:
Some learners may try to find multiplicative relationships among the original numbers, which can be misleading when the true pattern lies in the differences. Others might miscalculate one of the intermediate differences, which obscures the neat sequence of multiples of 12. A careful and systematic calculation of differences is essential to correctly identify and extend the underlying pattern.

Final Answer:
The missing number that completes the series is 133.