A series is given with one term missing. Choose the correct alternative from the given options that will complete the series: 6, 10, 18, 34, ?, 130

Introduction / Context:
This series uses differences that double at each step. The numbers 6, 10, 18, 34, ?, 130 grow increasingly quickly, and the pattern lies not in the terms themselves but in the differences between them. Recognising this structure is important for handling more advanced number series questions.

Given Data / Assumptions:
- Series: 6, 10, 18, 34, ?, 130.- One term between 34 and 130 is missing.- All terms are positive integers and increase rapidly.- Differences appear to double each time.

Concept / Approach:
The approach is to compute first differences and then examine whether these differences form a geometric sequence. If each difference is twice the previous one, we can generate the next difference by doubling again and use that to find the missing term. This is similar to working with exponential growth in the step sizes.

Step-by-Step Solution:
- From 6 to 10: 10 - 6 = 4.- From 10 to 18: 18 - 10 = 8.- From 18 to 34: 34 - 18 = 16.- The differences are 4, 8, 16, which double each time.- The next difference should be 16 * 2 = 32.- Therefore, the missing term after 34 is 34 + 32 = 66.- To reach 130, the next difference should double again: 32 * 2 = 64.- Check: 66 + 64 = 130, which matches the given final term.

Verification / Alternative check:
- With 66 inserted, the complete difference sequence is 4, 8, 16, 32, 64.- Each difference is exactly twice the previous one, confirming a geometric pattern in the increments.- No other candidate produces this clean doubling sequence.

Why Other Options Are Wrong:
- 52, 78, 94 and 82 lead to final differences that do not follow the doubling pattern from 4 to 64.- For example, if 78 were used, then 78 - 34 = 44 and 130 - 78 = 52, and the differences 4, 8, 16, 44, 52 do not maintain a consistent rule.

Common Pitfalls:
- Failing to compute all differences can cause candidates to miss the clear doubling trend.- Some might search for a complex formula for the main terms instead of noticing the simpler structure in the differences.- Quick guessing based on approximate position between 34 and 130, rather than pattern verification, often leads to wrong answers.

Final Answer:
The differences double at each step (4, 8, 16, 32, 64), so the missing term is 66.