Mathematical Group Analysis: Identifying the Property-Free Set

Mark in which group there is no property.

To determine which sequence has no consistent pattern or property, we will examine each of the given sequences for an arithmetic progression. An arithmetic progression is defined by a sequence in which each term after the first is obtained by adding a constant difference to the previous term.

• Option 1: $6, 5, 4, 3, 2, 1$ - The differences are $-1, -1, -1, -1, -1$. This follows an arithmetic progression with a common difference of $-1$.
• Option 2: $2, 1, 0, 1, 2, 3$ - The differences are $-1, -1, +1, +1, +1$. The differences are inconsistent across the entire sequence, suggesting this does not follow a simple arithmetic progression.
• Option 3: $55, 45, 35, 30, 20, 0$ - The differences are $-10, -10, -5, -10, -20$. The differences are inconsistent, indicating no uniform arithmetic progression property is present throughout.
• Option 4: $60, 50, 40, 30, 20, 10$ - The differences are $-10, -10, -10, -10, -10$. This sequence follows an arithmetic progression with a constant difference of $-10$.

From this analysis, Option 3 does not follow a consistent arithmetic progression or property, as the differences between terms are inconsistent.

Therefore, the sequence in which there is no property is $55, 45, 35, 30, 20, 0$.