Number Series — Find the Next Term Identify the next number in the sequence: 3, 12, 27, 48, 75, 108, ?

Introduction / Context:
This is a classic number series question where the pattern is based on successive differences. Recognizing difference patterns is a fundamental technique in quantitative aptitude and competitive exams.

Given Data / Assumptions:

• Series terms: 3, 12, 27, 48, 75, 108, ?
• We assume a deterministic arithmetic pattern in first or higher-order differences.

Concept / Approach:
Check consecutive differences. If differences themselves follow a clear progression (such as an arithmetic sequence), extend that progression to get the next difference, then add it to the last known term to obtain the answer.

Step-by-Step Solution:
Compute differences: 12 − 3 = 927 − 12 = 1548 − 27 = 2175 − 48 = 27108 − 75 = 33The difference sequence is: 9, 15, 21, 27, 33These increase by 6 each time: 9 + 6 = 15, 15 + 6 = 21, ... , 27 + 6 = 33Next difference = 33 + 6 = 39Next term = last term + next difference = 108 + 39 = 147

Verification / Alternative check:
Observe that differences form an arithmetic progression with common difference 6. Extending once yields 39, confirming 147 as consistent with the pattern.

Why Other Options Are Wrong:

• 162, 183, 192: None are equal to 108 + 39; they violate the established difference pattern.

Common Pitfalls:
Jumping to a multiplicative pattern or skipping the check on differences. Always test first differences (and, if needed, second differences) before trying more complex patterns.

Final Answer:
147