Complete the number series: 13, 19, 37, ?, 141, 243. Choose the number that should replace the question mark so that the series follows a consistent rule.

Introduction / Context:
The series 13, 19, 37, ?, 141, 243 involves increasing numbers and one missing term. The pattern in the differences between terms is not linear, but these differences themselves often follow a systematic growth such as quadratically increasing. We will analyse these differences and identify a rule that leads to the missing number.

Given Data / Assumptions:

• Series: 13, 19, 37, ?, 141, 243.
• Exactly one term is missing between 37 and 141.
• The rule is expected to be consistent and based on arithmetic differences.

Concept / Approach:
We compute the differences between consecutive known terms. When these differences are not constant, we then compute the second level differences between those differences. A smoothly increasing sequence of second differences (for example adding 8 each time) often indicates a quadratic pattern in the original series.

Step-by-Step Solution:
1. Let the missing term be N. The series is 13, 19, 37, N, 141, 243. 2. Compute known differences: 19 - 13 = 6. 37 - 19 = 18. 141 - N = ?. 243 - 141 = 102. 3. For a coherent pattern, we expect the sequence of differences to grow in a regular way. 4. Suppose the differences are 6, 18, 38, 66, 102. This is a standard pattern used in such questions. 5. Under this assumption, the unknown difference between 37 and N is 38, and between N and 141 is 66. 6. From 37, adding 38 gives the missing term: N = 37 + 38 = 75. 7. Check the next difference: 141 - 75 = 66, which fits the assumed pattern of differences. 8. Now confirm the second level differences: First level differences: 6, 18, 38, 66, 102. Second level differences: 18 - 6 = 12, 38 - 18 = 20, 66 - 38 = 28, 102 - 66 = 36. 9. These second level differences 12, 20, 28, 36 increase by 8 each time: 20 - 12 = 8, 28 - 20 = 8, 36 - 28 = 8. 10. This kind of uniform growth in second differences confirms a smooth quadratic type pattern, making 75 the correct missing term.

Verification / Alternative check:
Rebuild the full series using the approved differences: 13 + 6 = 19. 19 + 18 = 37. 37 + 38 = 75. 75 + 66 = 141. 141 + 102 = 243. The series 13, 19, 37, 75, 141, 243 now has first differences 6, 18, 38, 66, 102 and second differences that increment by 8, which is nicely consistent.

Why Other Options Are Wrong:

• 58, 67, 96: Substituting any of these values yields difference sequences that do not produce steadily increasing second differences by a constant step. The quadratic style pattern is lost, so these candidates do not fit.

Common Pitfalls:
Some learners stop at the first level of differences and conclude that there is no pattern. However, many reasoning series are designed so that the differences are themselves increasing in a structured way. Computing second level differences is a crucial strategy to reveal such patterns.

Final Answer:
The number that correctly completes the series is 75.