Mathematical Group Analysis: Identifying the Property-Free Set

Q: What exactly is an arithmetic progression?

An arithmetic progression is a sequence where you add the same number (called the common difference) to get from one term to the next. For example: 2, 5, 8, 11 has a common difference of +3.

Q: How do I find the differences between terms?

Subtract each term from the next one: second term - first term, then third term - second term, and so on. For sequence $6, 5, 4, 3, 2, 1$: $5-6=-1$, $4-5=-1$, etc.

Q: What if some differences are the same but not all?

If the differences aren't all the same, then it's not a true arithmetic progression! Look for sequences where differences are completely inconsistent, like $-10, -10, -5, -10, -20$.

Q: Can negative differences still make an arithmetic progression?

Absolutely! A sequence like $60, 50, 40, 30, 20, 10$ has differences of $-10$ throughout, making it a decreasing arithmetic progression.

Q: Why is the sequence 55,45,35,30,20,0 the answer?

Because its differences are $-10, -10, -5, -10, -20$ - completely inconsistent! Unlike the other sequences, this one has no uniform pattern or property.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:04 Let's identify the sequence that isn't a series.

00:08 First, let's find the pattern in each sequence.

00:12 In this sequence, the pattern is adding 1 each time.

00:16 For this sequence, the pattern is subtracting 1.

00:35 Notice, in this sequence, the pattern isn't consistent.

00:42 Here, the pattern is to add 10 every step.

00:48 And that's how we solve this question!

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Mark in which group there is no property.

2

Step-by-step solution

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 $55, 45, 35, 30, 20, 0$ .

3

Final Answer

^{55 , 45 , 35 , 30 , 20 , 0}

Key Points to Remember

Essential concepts to master this topic

Pattern Recognition: Calculate differences between consecutive terms to identify arithmetic progressions
Technique: For sequence 6,5,4,3,2,1: differences are -1,-1,-1,-1,-1 (constant)
Check: Inconsistent differences like -10,-10,-5,-10,-20 indicate no arithmetic pattern ✓

Common Mistakes

Avoid these frequent errors

Assuming all number sequences must have patterns
Don't assume every sequence follows an arithmetic progression = missing the correct answer! Some sequences are deliberately constructed without consistent properties. Always calculate all consecutive differences to verify if a pattern truly exists.

Practice Quiz

Test your knowledge with interactive questions

12 ☐ 10 ☐ 8 7 6 5 4 3 2 1

Which numbers are missing from the sequence so that the sequence has a term-to-term rule?

FAQ

Everything you need to know about this question

What exactly is an arithmetic progression?

+

An arithmetic progression is a sequence where you add the same number (called the common difference) to get from one term to the next. For example: 2, 5, 8, 11 has a common difference of +3.

How do I find the differences between terms?

+

Subtract each term from the next one: second term - first term, then third term - second term, and so on. For sequence $6, 5, 4, 3, 2, 1$ : $5-6=-1$ , $4-5=-1$ , etc.

What if some differences are the same but not all?

+

If the differences aren't all the same, then it's not a true arithmetic progression! Look for sequences where differences are completely inconsistent, like $-10, -10, -5, -10, -20$ .

Can negative differences still make an arithmetic progression?

+

Absolutely! A sequence like $60, 50, 40, 30, 20, 10$ has differences of $-10$ throughout, making it a decreasing arithmetic progression.

Why is the sequence 55,45,35,30,20,0 the answer?

+

Because its differences are $-10, -10, -5, -10, -20$ - completely inconsistent! Unlike the other sequences, this one has no uniform pattern or property.

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Mathematical Group Analysis: Identifying the Property-Free Set

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