Understanding the Relationship Between Numbers in Sets

The question asks us to identify a set of numbers that shares the same mathematical relationship as the given sets: (4, 6, 8) and (6, 8, 10). We need to find the underlying pattern or rule that connects the numbers within these sets, considering operations on the whole numbers.

Analyzing the Given Sets: (4, 6, 8) and (6, 8, 10)

Let's look at the relationship between the numbers in the first set (4, 6, 8):

• The difference between the second and first number is $6 - 4 = 2$.
• The difference between the third and second number is $8 - 6 = 2$.

The numbers have a constant difference between consecutive terms. This indicates that the numbers are in an arithmetic progression with a common difference of 2.

Now, let's analyze the second set (6, 8, 10):

• The difference between the second and first number is $8 - 6 = 2$.
• The difference between the third and second number is $10 - 8 = 2$.

Again, the numbers have a constant difference between consecutive terms, forming an arithmetic progression with a common difference of 2.

The common relationship observed in both given sets is that the numbers form an arithmetic progression. This means the difference between the second and first number is the same as the difference between the third and second number. Alternatively, the middle number is the average of the first and third number.

Checking the Options for the Same Number Relationship

We will now examine each option to see which set also forms an arithmetic progression.

Option 1: (2, 12, 8)

• Difference between second and first number: $12 - 2 = 10$.
• Difference between third and second number: $8 - 12 = -4$.

Since $10 \neq -4$, this set is not an arithmetic progression.

Option 2: (8, 12, 16)

• Difference between second and first number: $12 - 8 = 4$.
• Difference between third and second number: $16 - 12 = 4$.

Since $4 = 4$, the difference between consecutive numbers is constant. This set is an arithmetic progression with a common difference of 4.

Let's also check if the middle number is the average of the first and third: $(8 + 16) / 2 = 24 / 2 = 12$. The middle number is indeed the average.

This set follows the rule that the numbers are in an arithmetic progression, which is the same relationship found in the given sets.

Option 3: (5, 30, 10)

• Difference between second and first number: $30 - 5 = 25$.
• Difference between third and second number: $10 - 30 = -20$.

Since $25 \neq -20$, this set is not an arithmetic progression.

Option 4: (18, 18, 16)

• Difference between second and first number: $18 - 18 = 0$.
• Difference between third and second number: $16 - 18 = -2$.

Since $0 \neq -2$, this set is not an arithmetic progression.

Conclusion

The given sets (4, 6, 8) and (6, 8, 10) show that the numbers are in an arithmetic progression. By analyzing the options, we found that the set (8, 12, 16) also forms an arithmetic progression (with a common difference of 4). This confirms it shares the same type of relationship as the given sets.

Set Relation Analysis Revision Table

Additional Information on Number Series and Arithmetic Progressions

A sequence of numbers where the difference between consecutive terms is constant is called an arithmetic progression (AP). This constant difference is known as the common difference, denoted by 'd'.

In an arithmetic progression with three terms, say a, b, and c, the relationship is $b - a = c - b$. This can be rearranged to $2b = a + c$, which means the middle term is the average of the first and third terms: $b = (a+c)/2$.

The given sets (4, 6, 8) and (6, 8, 10) are examples of arithmetic progressions with a common difference of 2. The set (8, 12, 16) is an example of an arithmetic progression with a common difference of 4. Although the common difference is different, the fundamental relationship (being an arithmetic progression) is the same across these sets.