Find the next number in the series 5, 25, 61, 113, ... .

Introduction / Context:
This number series question involves a steadily increasing sequence where the gaps between terms grow according to a simple rule. The pattern in the first differences is the key to identifying the next number.

Given Data / Assumptions:

• Series: 5, 25, 61, 113, ...
• We need to find the next term after 113.
• The same rule generates each new term from the previous one.

Concept / Approach:
When dealing with a moderately fast growing sequence that does not appear purely geometric, we examine the differences between consecutive terms. If these differences follow a simple arithmetic progression, we can extend that to get the next difference and hence the missing term.

Step-by-Step Solution:
Step 1: Compute the differences: 25 − 5 = 20, 61 − 25 = 36, 113 − 61 = 52. Step 2: Look at the difference sequence: 20, 36, 52. Step 3: Compute the differences between these differences: 36 − 20 = 16, 52 − 36 = 16. Step 4: The second-level differences are constant at 16, which indicates that the first-level differences form an arithmetic progression with common difference 16. Step 5: Therefore the next difference should be 52 + 16 = 68. Step 6: Add this to the last known term: next term = 113 + 68 = 181.

Verification / Alternative check:
With the candidate next term 181, the extended series is 5, 25, 61, 113, 181. The differences are now 20, 36, 52, 68. The second-level differences remain 16, 16, 16, so the pattern is fully consistent. No other option maintains this simple second-level arithmetic progression.

Why Other Options Are Wrong:
If we choose 142, 156, or 179, the resulting difference from 113 does not equal 68, and the second-level differences no longer stay at 16. Each of those choices produces a break in the smooth progression of the first-level differences, so they cannot be correct.

Common Pitfalls:
A common mistake is to look only at the first differences and not consider second-level differences when the first ones do not form a simple pattern at first glance. Another error is to try to fit an ad hoc formula rather than exploiting the clean arithmetic progression present in the differences. Always remember that constant second differences are a strong clue to a quadratic type pattern.

Final Answer:
The next number in the series is 181.