Understanding Triad Relationships in Number Series

The question asks us to identify a triad (a set of three numbers) that shares the same relationship between its numbers as the two example triads provided: 15 - 14 - 13 and 17 - 21 - 25. We are explicitly told that operations should be performed on the whole numbers, not their individual digits.

Analyzing the Example Triads

Let's look closely at the relationship between the numbers in each example triad:

Example Triad 1: 15 - 14 - 13

• From 15 to 14: $14 = 15 - 1$
• From 14 to 13: $13 = 14 - 1$

In this triad, the numbers decrease by a constant value of 1. This is an arithmetic progression where the common difference is -1.

Example Triad 2: 17 - 21 - 25

• From 17 to 21: $21 = 17 + 4$
• From 21 to 25: $25 = 21 + 4$

In this triad, the numbers increase by a constant value of 4. This is an arithmetic progression where the common difference is +4.

Identifying the Common Relationship Pattern

Both example triads demonstrate a pattern where the difference between consecutive numbers is constant. This pattern is known as an arithmetic progression. An arithmetic progression is a sequence of numbers such that the difference between the consecutive terms is constant.

Let the three numbers in a triad be $a$, $b$, and $c$. If they form an arithmetic progression, then $b - a = c - b$, which means $2b = a + c$. The common difference is $d = b - a$ (or $d = c - b$).

Checking the Options for the Arithmetic Progression Pattern

Now, let's examine each option to see if its numbers form an arithmetic progression:

Option 1: 19 - 22 - 25

• Difference between 22 and 19: $22 - 19 = 3$
• Difference between 25 and 22: $25 - 22 = 3$

Since the difference between consecutive numbers is constant (3), this triad forms an arithmetic progression.

Option 2: 34 - 12 - 28

• Difference between 12 and 34: $12 - 34 = -22$
• Difference between 28 and 12: $28 - 12 = 16$

The differences (-22 and 16) are not constant. This triad does not form an arithmetic progression.

Option 3: 38 - 28 - 10

• Difference between 28 and 38: $28 - 38 = -10$
• Difference between 10 and 28: $10 - 28 = -18$

The differences (-10 and -18) are not constant. This triad does not form an arithmetic progression.

Option 4: 23 - 42 - 46

• Difference between 42 and 23: $42 - 23 = 19$
• Difference between 46 and 42: $46 - 42 = 4$

The differences (19 and 4) are not constant. This triad does not form an arithmetic progression.

Conclusion

Only Option 1, 19 - 22 - 25, exhibits the same underlying relationship as the example triads (15 - 14 - 13 and 17 - 21 - 25), which is forming an arithmetic progression. The common difference in Option 1 is 3, while the common differences in the examples are -1 and 4 respectively. The question asks for the numbers to be related in the 'same way', and the shared 'way' between the examples is being an arithmetic sequence.

Revision Table: Number Triad Relationships

Additional Information: Types of Number Series Patterns

Beyond arithmetic progressions, number series can follow many other patterns. Recognizing these patterns is key to solving such reasoning questions.

• Geometric Progression: Each term is found by multiplying the previous term by a constant value (the common ratio). E.g., 2, 4, 8, 16 (common ratio 2).
• Fibonacci Sequence: Each term is the sum of the two preceding ones. E.g., 0, 1, 1, 2, 3, 5, 8.
• Square Numbers: 1, 4, 9, 16, 25 ... ($n^2$)
• Cube Numbers: 1, 8, 27, 64, 125 ... ($n^3$)
• Differences Series: The differences between consecutive terms follow a recognizable pattern (e.g., an arithmetic progression themselves).
• Mixed Series: Combinations of different patterns or operations.

For triad problems like this, identifying the pattern common to the examples is the crucial first step before testing the options.