Choose the correct alternative that will continue the same pattern and replace x in the series: 6, 18, 42, 78, x.

Introduction / Context:
This is a standard number series question where the differences between consecutive terms are not constant but increase in a systematic way. You are required to determine the next number in the sequence by identifying how these differences grow. Problems like this are common in reasoning sections because they test your ability to recognise simple arithmetic progressions at the level of differences rather than the original terms.

Given Data / Assumptions:

• Series: 6, 18, 42, 78, x.
• Exactly one term x is missing at the end.
• The rule is expected to be straightforward, based on differences between terms.

Concept / Approach:
When the numbers increase fairly quickly and not in a fixed step, the first thing to check is the difference between consecutive terms. If those differences themselves form a simple arithmetic sequence, then you can extend that sequence to find the next difference and hence the missing term. This approach uses first differences and sometimes second differences if needed.

Step-by-Step Solution:
Step 1: Compute the successive differences. 18 - 6 = 12. 42 - 18 = 24. 78 - 42 = 36. Step 2: Collect these differences: 12, 24, 36. Step 3: Observe that the differences form a sequence where each step increases by 12: 12, 24, 36. Step 4: The next difference in this pattern should be 36 + 12 = 48. Step 5: Add this difference to the last known term of the original series: 78 + 48 = 126.

Verification / Alternative check:
Write the extended series with the found value: 6, 18, 42, 78, 126. Now recompute: 18 - 6 = 12, 42 - 18 = 24, 78 - 42 = 36, 126 - 78 = 48. The first differences 12, 24, 36, 48 clearly form an arithmetic progression with common difference 12. This symmetry strongly confirms that 126 is the correct next term.

Why Other Options Are Wrong:
Values like 84, 96, or 108 would produce difference sets that no longer follow the pattern of adding 12 each time. For example, if x were 96, then the last difference would be 96 - 78 = 18, which does not fit into the difference sequence 12, 24, 36. Since the question is built on a clear increasing difference pattern, any option that breaks this structure must be rejected.

Common Pitfalls:
One frequent mistake is to search for multiplicative relationships such as ratios between terms when a simple additive pattern is sufficient. Another is to guess based only on rough growth without checking the exact difference sequence. Always calculate and examine differences carefully, especially in series questions where numbers grow at a changing rate.

Final Answer:
The value of x that correctly completes the series is 126.