For the number sequence 3, 5, 13, 43, 167, identify the odd man out by analysing the recursive pattern linking each term to its position in the series.

Introduction / Context:
This odd man out problem involves the series 3, 5, 13, 43, 167. At first glance all numbers look somewhat prime like, so simple tests like checking for prime numbers do not immediately give a unique outlier. Instead, the hidden pattern connects each term to the previous one using its position in the series.

Given Data / Assumptions:
The terms are:

• a(1) = 3
• a(2) = 5
• a(3) = 13
• a(4) = 43
• a(5) = 167

We assume that for most terms there is a neat recurrence relation involving multiplication by the term position and then adding a small integer that depends on that position.

Concept / Approach:
A key idea is to test whether a(n+1) can be written as a(n) * n + (n + 1), where n is the index of a(n) starting from 1. If this works for all but one term, then that exceptional term is the odd man out. This strategy is common in series where numbers grow faster than a simple arithmetic progression but not as fast as a pure geometric progression.

Step-by-Step Solution:
Step 1: From a(1) = 3 to a(2) = 5. At n = 1, compute 3 * 1 + (1 + 1) = 3 * 1 + 2 = 5, which matches a(2).Step 2: From a(2) = 5 to a(3) = 13. At n = 2, compute 5 * 2 + (2 + 1) = 10 + 3 = 13, which matches a(3).Step 3: From a(3) = 13 to a(4) = 43. At n = 3, compute 13 * 3 + (3 + 1) = 39 + 4 = 43, which matches a(4).Step 4: From a(4) = 43 to the next term. At n = 4, the rule gives a(5) should be 43 * 4 + (4 + 1) = 172 + 5 = 177.Step 5: However, the given fifth term is 167, not 177. Therefore, 167 does not satisfy the same recurrence relation.

Verification / Alternative check:
We can propose the intended pattern as a(1) = 3 and, for n ≥ 1, a(n + 1) = a(n) * n + (n + 1). This formula correctly generates 5, 13 and 43 from 3, 5 and 13 respectively. Extending it gives 177 as the next value after 43. Because all earlier steps fit perfectly, the most consistent interpretation is that the correct fifth term should have been 177 and 167 is an incorrect outlier.

Why Other Options Are Wrong:
The terms 5, 13 and 43 are all exactly produced by the recurrence a(n + 1) = a(n) * n + (n + 1). Removing any of them would break the recurrence earlier in the sequence. Only the last term, 167, fails to satisfy the pattern, making it the natural odd man out.

Common Pitfalls:
It is tempting to classify numbers simply based on primality or to search for simple difference patterns. Since all the given numbers are prime, that approach does not distinguish any one element. Recognising a position dependent recursion is the key here. Always consider the possibility that the index n itself is part of the rule that generates the series.

Final Answer:
The number that does not follow the established recursive pattern is 167.