In the series 5, 7, ?, 25, 45, 75, what number should replace the question mark to maintain the pattern?

Introduction / Context:
This is a number series question where you must find the missing term. The pattern is based on the differences between consecutive terms and how those differences themselves change. Such problems test your ability to identify second level patterns and not just direct increments.

Given Data / Assumptions:

• Given series: 5, 7, ?, 25, 45, 75.
• Exactly one number is missing.
• We assume the sequence is constructed in a consistent way based on simple arithmetic operations.

Concept / Approach:
To find the missing term, we look at the pattern of differences. We know the first term is 5 and the second is 7. The unknown third term must fit smoothly between 7 and 25 and then between 25 and 45 and 75. If the differences follow a regular rule like an arithmetic progression, we can use this to determine the missing number.

Step-by-Step Solution:
Let the missing term be x. Then the series becomes 5, 7, x, 25, 45, 75.Compute the known differences: 25 - x, 45 - 25 = 20, 75 - 45 = 30.We already know 7 - 5 = 2 and x - 7 is another difference we must determine.Look for a pattern where the differences themselves follow an arithmetic progression. Suppose the differences are 2, 6, 12, 20, 30.Check this: 5 + 2 = 7, 7 + 6 = 13, 13 + 12 = 25, 25 + 20 = 45, 45 + 30 = 75.This works perfectly and shows that the missing term must be 13.

Verification / Alternative check:
With the completed series 5, 7, 13, 25, 45, 75, the differences are 2, 6, 12, 20 and 30. The second differences are 4, 6, 8 and 10, which increase by 2 each time. This smooth and systematic behavior strongly indicates that 13 is the correct missing term. No other candidate will produce such a neat pattern in both first and second differences.

Why Other Options Are Wrong:
11, 15, 17 and 19 all lead to irregular or messy patterns of differences that do not form a clear arithmetic progression or a simple structure in the second differences. For example, using 11 creates differences 2, 4, 14, 20 and 30, where second differences are not consistently increasing by a fixed amount.

Common Pitfalls:
Many test takers only check the visible part after the missing term and try to fit 25, 45 and 75 with the wrong value, which leads to random guessing. A systematic way is to express the unknown as x, compute differences and search for a reasonable pattern such as steadily increasing second differences. This structured reasoning avoids guesswork and improves accuracy.

Final Answer:
The missing number in the series is 13.