In the series 3, 12, 39, 120, 364, identify the odd man out by detecting the underlying multiplicative plus constant pattern.

Introduction / Context:
The series 3, 12, 39, 120, 364 contains one term that does not fit the simple pattern followed by the others. The numbers grow quickly, suggesting a rule that involves multiplication by an increasing integer and then adding a constant. Recognising and checking this type of pattern is a common requirement in number series questions.

Given Data / Assumptions:
The terms are:

• 3
• 12
• 39
• 120
• 364

We assume that there is a consistent recurrence relation of the form a(n+1) = a(n) * k + c, where k and c depend in a simple way on the position n, and that one of the terms has been misgenerated or misprinted.

Concept / Approach:
We test whether each term can be obtained by multiplying the previous term by 3 and then adding 3. If this rule works for all but one term, that exceptional term is the odd man out. This pattern is very typical: multiply by 3, then add a small constant, repeated across the series.

Step-by-Step Solution:
Step 1: From 3 to 12. Check 3 * 3 + 3 = 9 + 3 = 12, which matches.Step 2: From 12 to 39. Check 12 * 3 + 3 = 36 + 3 = 39, which matches.Step 3: From 39 to 120. Check 39 * 3 + 3 = 117 + 3 = 120, which matches.Step 4: From 120 to the next term according to the same rule. Compute 120 * 3 + 3 = 360 + 3 = 363.Step 5: The given next term is 364, not 363. Hence 364 does not satisfy the established rule a(n+1) = 3 * a(n) + 3.

Verification / Alternative check:
If we assume the intended sequence was 3, 12, 39, 120, 363, then every term after the first is exactly three times the previous term plus 3. There are no contradictions or irregularities in this corrected sequence, which strongly indicates that 364 is a misfit and should have been 363 to keep the pattern consistent.

Why Other Options Are Wrong:
The terms 3, 12, 39 and 120 all strictly follow the rule a(n+1) = 3 * a(n) + 3. Removing any of them as the odd man out would break the pattern earlier in the series, while still leaving 364 as an outlier. Therefore, they must be considered correct members of the sequence, and 364 is the only reasonable candidate for removal.

Common Pitfalls:
Some students try to analyse first differences or ratios alone and may overlook the simple linear recurrence involving multiplication and a constant addition. When numbers grow by about a factor of 3, it is very helpful to test a rule like a(n+1) = 3 * a(n) ± c before resorting to more complicated approaches.

Final Answer:
The number that does not fit the pattern a(n+1) = 3 * a(n) + 3 is 364.