Understanding the Number Pattern

The question asks us to analyze a given pattern of numbers presented in sets of three and determine the number that should replace the question mark. The pattern is given as:

(2, 2, 4)

(1, 6, 6)

(7, 2, ?)

We are instructed to use operations on the whole numbers themselves, not their individual digits.

Analyzing the Pattern in the Given Sets

Let's look at the relationship between the numbers in the first two sets to find a consistent pattern. Let the three numbers in each set be represented by $a$, $b$, and $c$. So, a set is $(a, b, c)$.

Examining the First Set: (2, 2, 4)

Here, $a=2$, $b=2$, and $c=4$. Let's test some simple mathematical operations:

• Addition: $a + b = 2 + 2 = 4$. This equals $c$.
• Multiplication: $a \times b = 2 \times 2 = 4$. This also equals $c$.
• Other operations like subtraction ($a-b=0 \neq 4$), division ($a/b=1 \neq 4$), or squares ($a^2=4$, $b^2=4$) might match, but we need a pattern involving all three numbers, or at least a consistent relationship between the first two and the third.

Both addition and multiplication work for the first set.

Examining the Second Set: (1, 6, 6)

Here, $a=1$, $b=6$, and $c=6$. Let's test the potential patterns we found from the first set:

• Addition: $a + b = 1 + 6 = 7$. This is not equal to $c$ (which is 6). So, simple addition ($a+b=c$) is not the pattern.
• Multiplication: $a \times b = 1 \times 6 = 6$. This equals $c$.

The multiplication pattern ($a \times b = c$) works for both the first and the second set.

Applying the Pattern to the Third Set

The consistent pattern observed across the first two sets is that the product of the first two numbers equals the third number ($a \times b = c$). Let's apply this pattern to the third set: (7, 2, ?).

Here, $a=7$ and $b=2$. We need to find $c$, which is the missing number represented by the question mark (?).

Using the pattern:

$c = a \times b$

$c = 7 \times 2$

$c = 14$

So, the missing number is 14.

Verifying the Result with Options

The calculated missing number is 14. Let's check the given options:

1. 14
2. 2
3. 7
4. 12

The number 14 matches Option 1.

Detailed Solution Steps

1. Identify the three numbers in each set: $(a, b, c)$.
2. Analyze the relationship between $a$, $b$, and $c$ in the first set (2, 2, 4). Possible patterns include $a+b=c$ or $a \times b = c$.
3. Test the potential patterns on the second set (1, 6, 6). $1+6=7 \neq 6$, so $a+b=c$ is not the pattern. $1 \times 6 = 6$, so $a \times b = c$ is a valid pattern so far.
4. Confirm the pattern $a \times b = c$ holds for the first set again: $2 \times 2 = 4$. Yes, it holds.
5. Apply the pattern $a \times b = c$ to the third set (7, 2, ?). Let the missing number be $x$. So, $7 \times 2 = x$.
6. Calculate the value of $x$: $x = 14$.
7. The missing number is 14.
8. Match the result with the given options.

The number that can replace the question mark (?) is 14.

Revision Table: Key Concepts

Additional Information: Types of Number Patterns

Number patterns can appear in various forms, often tested in logical reasoning or quantitative aptitude sections. Some common types include:

• Arithmetic Sequences: Each term is obtained by adding a constant value to the previous term (e.g., 2, 4, 6, 8... adding 2 each time).
• Geometric Sequences: Each term is obtained by multiplying the previous term by a constant value (e.g., 3, 9, 27, 81... multiplying by 3 each time).
• Fibonacci Sequence: Each term is the sum of the two preceding terms (e.g., 0, 1, 1, 2, 3, 5, 8...).
• Square Numbers: Sequence of numbers that are perfect squares (e.g., 1, 4, 9, 16, 25...).
• Cube Numbers: Sequence of numbers that are perfect cubes (e.g., 1, 8, 27, 64...).
• Mixed Operations: Patterns involving a combination of operations or operations between terms in a set, as seen in this problem ($a \times b = c$).

Solving pattern problems often involves carefully observing the given examples, testing different mathematical relationships, and verifying the found rule across all provided data points before applying it to find the missing element.