Solving the Number Triad Pattern

The question asks us to complete a sequence of numbers presented as triads. A triad is a set of three numbers. The sequence is given as: 12 20 28 21 19 ? 31 21 11.

We can group these numbers into three distinct triads:

• Triad 1: (12, 20, 28)
• Triad 2: (21, 19, ?)
• Triad 3: (31, 21, 11)

To find the missing number in the second triad, we need to identify the pattern or relationship that connects the numbers within each triad. Let's examine the first and third triads for a consistent pattern.

Analyzing Triad 1: (12, 20, 28)

Let's look at the relationship between these numbers. The number 20 is exactly in the middle of 12 and 28 if we consider their position on the number line or in terms of arithmetic progression.

• Difference between the second and first number: $20 - 12 = 8$
• Difference between the third and second number: $28 - 20 = 8$

This shows a constant difference of 8. Another way to look at this is that the middle number is the average of the first and third numbers:

$\frac{12 + 28}{2} = \frac{40}{2} = 20$

The middle number (20) is the average of the first (12) and third (28) numbers in the first triad.

Analyzing Triad 3: (31, 21, 11)

Let's check if the same pattern holds for the third triad.

• Difference between the first and second number: $31 - 21 = 10$
• Difference between the second and third number: $21 - 11 = 10$

Here, the constant difference is 10. Again, let's check if the middle number is the average of the first and third numbers:

$\frac{31 + 11}{2} = \frac{42}{2} = 21$

The middle number (21) is indeed the average of the first (31) and third (11) numbers in the third triad.

Applying the Pattern to Triad 2: (21, 19, ?)

Based on the pattern observed in the first and third triads, the middle number in a triad is the average of the first and third numbers. In the second triad (21, 19, ?), the first number is 21, the middle number is 19, and the third number is the unknown, let's call it $x$.

Using the pattern, the middle number (19) must be the average of the first (21) and third ($x$) numbers:

$19 = \frac{21 + x}{2}$

Now, we need to solve this equation for $x$ to find the missing number.

Solving for the Missing Number

Step 1: Multiply both sides of the equation by 2.

$19 \times 2 = \frac{21 + x}{2} \times 2$

$38 = 21 + x$

Step 2: Subtract 21 from both sides of the equation to isolate $x$.

$38 - 21 = 21 + x - 21$

$17 = x$

So, the missing number in the triad (21, 19, ?) is 17.

Verifying the Triad (21, 19, 17)

Let's check if 19 is the average of 21 and 17:

$\frac{21 + 17}{2} = \frac{38}{2} = 19$

This confirms that 19 is the average of 21 and 17, so the pattern holds for this triad as well.

Conclusion

The missing number in the sequence is 17. This corresponds to one of the given options.

Revision Table: Number Triad Patterns

Understanding patterns is key to solving number sequence and triad problems. Common patterns include arithmetic progression (constant difference), geometric progression (constant ratio), squaring/cubing numbers, Fibonacci sequence, or combinations of these. In this specific triad puzzle, the pattern was that the middle number is the arithmetic mean (average) of the first and third numbers.

Additional Information: Arithmetic Mean

The arithmetic mean, or average, of a set of numbers is the sum of the numbers divided by the count of the numbers. For two numbers, say $a$ and $b$, their arithmetic mean is $\frac{a+b}{2}$. In an arithmetic progression of three terms, say $a, b, c$, the middle term $b$ is the arithmetic mean of the first and third terms, $a$ and $c$. That is, $b = \frac{a+c}{2}$, which also means $2b = a+c$. This relationship is fundamental in solving problems involving arithmetic sequences.