Pattern Identification in Number Sets

The question presents two sets of numbers, each following a specific sequence of mathematical operations. The rule is that operations are applied to the whole number, not its digits.

Set 1 Analysis: 6 - 36 - 72 - 144

• The first number is 6.
• The second number, 36, is obtained by squaring the first number: \( 6 \times 6 = 36 \).
• The third number, 72, is obtained by doubling the second number: \( 36 \times 2 = 72 \).
• The fourth number, 144, is obtained by doubling the third number: \( 72 \times 2 = 144 \).

Set 2 Analysis: 7 - 49 - 98 - 196

• The first number is 7.
• The second number, 49, is obtained by squaring the first number: \( 7 \times 7 = 49 \).
• The third number, 98, is obtained by doubling the second number: \( 49 \times 2 = 98 \).
• The fourth number, 196, is obtained by doubling the third number: \( 98 \times 2 = 196 \).

Identified Mathematical Operations

Both sets follow the same pattern:

1. Square the first number to get the second number.
2. Double the second number to get the third number.
3. Double the third number to get the fourth number.

Evaluating Options Against the Pattern

We need to find the option that adheres to the \( x \rightarrow x^2 \rightarrow (x^2) \times 2 \rightarrow ((x^2) \times 2) \times 2 \) pattern.

Checking Option 1: 12 - 144 - 288 - 576

• First number: 12
• Square of the first number: \( 12 \times 12 = 144 \). (Matches the second number)
• Double the second number: \( 144 \times 2 = 288 \). (Matches the third number)
• Double the third number: \( 288 \times 2 = 576 \). (Matches the fourth number)

This option perfectly follows the identified pattern.

Checking Other Options

• Option 2: 5 - 25 - 50 - 110. While \( 5 \times 5 = 25 \) and \( 25 \times 2 = 50 \) are correct, \( 50 \times 2 \) should be 100, not 110.
• Option 3: 15 - 225 - 625 - 676. While \( 15 \times 15 = 225 \) is correct, \( 225 \times 2 \) should be 450, not 625.
• Option 4: 8 - 64 - 128 - 226. While \( 8 \times 8 = 64 \) and \( 64 \times 2 = 128 \) are correct, \( 128 \times 2 \) should be 256, not 226.

Conclusion

Only Option 1 follows the exact sequence of mathematical operations (square, then double twice) observed in the given sets.