A series is given with one term missing. Choose the correct alternative from the given options that will complete the series: 2, 10, 30, 68, ?

Introduction / Context:
This number series grows quickly, and the jumps between terms themselves change in a regular way. Such questions typically involve second differences that form an arithmetic progression. The series is 2, 10, 30, 68, ?, and we must deduce the next term from the pattern hidden in the differences.

Given Data / Assumptions:
- Series terms: 2, 10, 30, 68, ?- One term, the fifth, is missing.- The series is strictly increasing and accelerates as it grows.- The pattern is likely based on changing differences rather than a single multiplication factor.

Concept / Approach:
When the growth between terms increases, it is helpful to compute first differences and then second differences. If second differences follow a simple arithmetic rule (for example, increasing by a fixed amount), we can extend that rule forward. This method is similar to working with quadratic sequences in algebra.

Step-by-Step Solution:
- Compute first differences: • From 2 to 10: 10 - 2 = 8. • From 10 to 30: 30 - 10 = 20. • From 30 to 68: 68 - 30 = 38.- First differences are 8, 20, 38.- Compute second differences: • 20 - 8 = 12. • 38 - 20 = 18.- Second differences are 12 and 18, which increase by 6.- Continue this pattern: next second difference = 18 + 6 = 24.- Therefore the next first difference = 38 + 24 = 62.- Add this to the last known term: 68 + 62 = 130.- So the missing term is 130.

Verification / Alternative check:
- With the missing term as 130, the first differences become 8, 20, 38, 62.- The second differences are then 12, 18, 24, which clearly form an arithmetic progression with common difference 6.- This confirms that 130 is consistent with the underlying rule.

Why Other Options Are Wrong:
- 135, 140, 120 and 150 do not produce first and second differences that continue the sequence 12, 18, 24 in the second differences.- For example, using 135 would give the last first difference as 67, leading to a second difference of 29, which breaks the pattern.

Common Pitfalls:
- Some candidates try to fit a single multiplication factor like times 3 or times 2 plus a constant, which does not work cleanly here.- Failing to compute second differences causes the regularity in the pattern to be missed.- Random guessing based only on approximate growth often leads to values that do not preserve the structured second differences.

Final Answer:
The second differences form an arithmetic progression 12, 18, 24, so the next term in the series is 130.