In this three-number group odd-man-out question, select the group whose numbers do not follow the same linear relationship.

Introduction / Context:
This is a number-group classification question in which each option is an ordered triple of numbers. In such reasoning questions, the second and third numbers are usually related to the first by a simple arithmetic rule. Three groups follow the same pair of rules, and one group breaks that pattern. Our task is to identify the group that does not respect the same linear relationships.

Given Data / Assumptions:

• Groups given: (8, 17, 33), (11, 23, 45), (13, 27, 51) and (17, 35, 69).
• Each triple is written as (first, second, third).
• We are looking for a consistent arithmetic relation that three triples share but one does not.

Concept / Approach:
The pattern here involves expressing the second and third numbers in terms of the first. A very natural attempt is to check whether the second number is approximately double the first and the third number approximately double the second. Specifically, we test simple linear forms such as second = 2 * first + constant and third = 2 * second - constant. When this form uses the same constants for three groups but fails for one group, that group is the odd-man-out.

Step-by-Step Solution:
Step 1: Analyse (8, 17, 33). Compare the second and first numbers. We have 2 * 8 + 1 = 17. So second = 2 * first + 1. Compare the third and second numbers. We have 2 * 17 - 1 = 33. So third = 2 * second - 1. Step 2: Analyse (11, 23, 45). Compute 2 * 11 + 1 = 23, so second = 2 * first + 1 again. Now compute 2 * 23 - 1 = 45, so third = 2 * second - 1. This triple follows the same pair of formulas. Step 3: Analyse (17, 35, 69). Compute 2 * 17 + 1 = 35, so the second number again obeys second = 2 * first + 1. Next compute 2 * 35 - 1 = 69, which matches the third number. Thus, this triple also follows the same rules. Step 4: Analyse (13, 27, 51). Compute 2 * 13 + 1 = 27, so the second number fits second = 2 * first + 1. Now compute 2 * 27 - 1 = 53, but the third number given is 51. Therefore, third ≠ 2 * second - 1 in this triple. Step 5: Summarise. Three triples satisfy two common linear relations: second = 2 * first + 1 and third = 2 * second - 1. The triple (13, 27, 51) satisfies only the first relation but fails the second. Step 6: Conclude that (13, 27, 51) is the odd group because its third term does not follow the same linear rule that applies to the other groups.

Verification / Alternative check:
We may also express the third term directly in terms of the first term to see the pattern more clearly. For (8, 17, 33), using third = 4 * first + 1 gives 4 * 8 + 1 = 33. For (11, 23, 45), 4 * 11 + 1 = 45. For (17, 35, 69), 4 * 17 + 1 = 69. So for three groups, third = 4 * first + 1. For the triple (13, 27, 51), we get 4 * 13 + 1 = 53, while the third term is 51. This again shows that the third term does not match the same formula, independently confirming our earlier conclusion.

Why Other Options Are Wrong:
(8, 17, 33) is not odd because it perfectly fits both relations: second = 2 * first + 1 and third = 2 * second - 1. (11, 23, 45) also follows both rules. (17, 35, 69) continues the pattern exactly. Since all three triples are structurally identical under the same set of formulas, they cannot be the odd-man-out. Only (13, 27, 51) breaks the relationship for the third term.

Common Pitfalls:
Some learners only check the relation between the first and second numbers and stop as soon as they see that all groups satisfy second = 2 * first + 1. If they do not verify the relation between the second and third numbers, they might think there is no odd-man-out at all. A second pitfall is trying to guess patterns based solely on differences rather than testing simple linear formulas systematically. The best practice is to test all parts of the triple so that you do not miss an inconsistency like the one in this question.

Final Answer:
The odd group is (13, 27, 51), because in this group the third number does not satisfy the rule third = 2 * second - 1, while all the other groups do follow this linear pattern exactly.