Consider the number series: 4, 18, ?, 100, 180, 294, 448 Find the missing term that should replace the question mark so that the pattern in the series is maintained.

Introduction / Context:
This is a number series question in which one term is missing. The goal is to detect the hidden pattern connecting consecutive terms and then use that pattern to determine the unknown value. Here the pattern is not based on simple multiplication or addition, so we need to examine the differences between terms more carefully.

Given Data / Assumptions:

• Series: 4, 18, ?, 100, 180, 294, 448.
• Exactly one term (the third) is missing.
• The given sequence is assumed to follow a consistent mathematical pattern.
• We must choose the missing term from the options 48, 50, 58 and 60.

Concept / Approach:
When the pattern is not obvious from the numbers themselves, it is useful to look at first differences (successive subtractions) and then second differences. If these differences show a regular structure, they can guide us to the correct missing term. In this series, a cubic-type pattern in the differences appears when the correct missing value is used.

Step-by-Step Solution:
Step 1: Assume the missing term is T and write the series as 4, 18, T, 100, 180, 294, 448. Step 2: If we try T = 48, the sequence becomes 4, 18, 48, 100, 180, 294, 448. Step 3: Compute the first differences: 18 - 4 = 14, 48 - 18 = 30, 100 - 48 = 52, 180 - 100 = 80, 294 - 180 = 114, 448 - 294 = 154. Step 4: Now compute the second differences: 30 - 14 = 16, 52 - 30 = 22, 80 - 52 = 28, 114 - 80 = 34, 154 - 114 = 40. Step 5: Observe that these second differences form an arithmetic sequence themselves: 16, 22, 28, 34, 40 increase by 6 each time. Step 6: This smooth second-difference pattern strongly indicates that T = 48 is correct and that the series is generated by a cubic polynomial relation.

Verification / Alternative check:
If we try T = 50 or 58 or 60, the first and second differences do not follow such a neat and steadily increasing pattern. The differences become irregular, which is unlikely for a well-set exam series. Thus only 48 leads to a consistent and elegant structure in the differences.

Why Other Options Are Wrong:
Choosing 50, 58 or 60 breaks the progression of second differences, making them jump in an inconsistent way. Since such irregularity rarely indicates an intended exam pattern, these values are not acceptable as the missing term.

Common Pitfalls:
A common mistake is to search only for simple multiplication or addition rules between consecutive terms, and to give up when that fails. Some test-takers also stop after checking only first differences. For more complex series, examining second or even higher-order differences is often necessary to reveal the underlying rule.

Final Answer:
The missing term that maintains a smooth pattern in the differences is 48, so option (a) is the correct answer.