In this three-number group odd-man-out question, select the group whose numbers do not form an arithmetic progression with the same common difference.

Introduction / Context:
This is a number-sequence classification question involving groups of three numbers. In three of the groups, the numbers form an arithmetic progression with the same common difference. In one group, that pattern breaks because the third number does not follow the same step size. Recognising this pattern allows us to identify the odd-man-out quickly.

Given Data / Assumptions:

• Groups given: (7, 14, 21), (5, 12, 19), (3, 10, 17) and (9, 16, 35).
• Each triple is ordered as (first, second, third).
• We are looking at the differences between consecutive terms in each triple.

Concept / Approach:
An arithmetic progression is a sequence of numbers where the difference between any two consecutive terms is constant. In this question, we check whether the second minus the first and the third minus the second give the same common difference within a triple, and also whether this common difference is the same across triples. We will see that three triples share a common difference of 7 throughout, while one triple deviates at the last step.

Step-by-Step Solution:
Step 1: Analyse (7, 14, 21). The differences are 14 - 7 = 7 and 21 - 14 = 7. This group forms an arithmetic progression with common difference 7. Step 2: Analyse (5, 12, 19). The differences are 12 - 5 = 7 and 19 - 12 = 7. This triple also has common difference 7. Step 3: Analyse (3, 10, 17). The differences are 10 - 3 = 7 and 17 - 10 = 7. This group again is an arithmetic progression with common difference 7. Step 4: Analyse (9, 16, 35). The differences are 16 - 9 = 7 and 35 - 16 = 19. The first step is 7, but the second step is 19, which is not equal to 7. So the numbers here do not form an arithmetic progression with constant difference. Step 5: Summarise. Three groups form arithmetic progressions with the same common difference of 7, while one group has a different second step. Step 6: Conclude that (9, 16, 35) is the odd group because it does not maintain the constant common difference of 7.

Verification / Alternative check:
We can also express the third term directly in terms of the first using the formula for an arithmetic progression: third = first + 2 * common difference. In the first three groups, third = first + 14. For example, 7 + 14 = 21, 5 + 14 = 19 and 3 + 14 = 17. In the last group, 9 + 14 = 23, but the third term is 35, not 23. This confirms that (9, 16, 35) does not follow the same progression and supports our conclusion from the difference method.

Why Other Options Are Wrong:
(7, 14, 21) is not odd because it is a clean arithmetic progression with common difference 7. (5, 12, 19) also has the same difference structure. (3, 10, 17) continues that pattern. Since these three triples all share the same arithmetic progression form with identical common difference, none of them can be the odd-man-out. Only (9, 16, 35) fails that rule by using a different second difference.

Common Pitfalls:
One common mistake is to just observe that 35 is far larger than the others and assume that size alone makes the triple odd, without actually verifying the pattern. While size hints that something may be different, the correct reasoning still depends on checking the differences systematically. Another error is to miscalculate the differences. To avoid such mistakes, always compute both steps carefully and compare them, rather than relying on approximate visual impressions.

Final Answer:
The odd group of numbers is (9, 16, 35), because it does not form an arithmetic progression with common difference 7, whereas all other groups do.