In the series 1, 6, 27, 124, 650, which number is the odd man out that does not follow the same rule as the others?

Introduction / Context:
This question asks you to identify the odd term in a series built using a mix of multiplication and addition. It is based on a pattern where each term is constructed from the previous one using a simple rule involving the position index. Spotting such a pattern is a common challenge in number series questions.

Given Data / Assumptions:

• Sequence: 1, 6, 27, 124, 650.
• We assume that from the first few terms we can infer a rule.
• Exactly one term in the list breaks that rule.

Concept / Approach:
We look for a relation between each term and the next one. A useful idea is that the multiplier and the added quantity might depend on the step number. By comparing how 1 becomes 6, then 6 becomes 27 and 27 becomes 124, we can attempt to generalize the pattern and then test whether 650 fits or violates it.

Step-by-Step Solution:
From 1 to 6: 1 * 2 + 2^2 = 2 + 4 = 6.From 6 to 27: 6 * 3 + 3^2 = 18 + 9 = 27.From 27 to 124: 27 * 4 + 4^2 = 108 + 16 = 124.We see a clear rule: if Tn is the current term at step n, the next term is Tn * (n + 1) + (n + 1)^2.Following this rule, from 124 to the next term we should have: 124 * 5 + 5^2 = 620 + 25 = 645.The term in the series is 650, which does not match the required value 645. Therefore, 650 is inconsistent with the pattern.

Verification / Alternative check:
If we write the correct sequence according to the discovered rule, it becomes 1, 6, 27, 124, 645, and so on. All transitions follow the same formula using multipliers and squares of consecutive integers. Only 650 deviates from this regularity by being 5 units too large. This confirms it as the wrong term.

Why Other Options Are Wrong:
1 is the starting value and fits naturally into the pattern.

6, 27 and 124 all satisfy the relation Tn+1 = Tn * (n + 1) + (n + 1)^2 exactly.

Removing any one of these would destroy the internal consistency of the early part of the series, while the pattern up to 124 is perfectly smooth.

Common Pitfalls:
Some candidates may miscalculate one of the intermediate steps and conclude that an earlier term is wrong. Others might look only at rough magnitude growth and not check exact arithmetic. It is important to verify each step precisely and test whether a single formula works for all transitions. Once that formula is found, identifying the one term that breaks it becomes straightforward.

Final Answer:
The odd man out in the sequence is 650.