Identifying the Number Operations Pattern

The problem presents two sets of numbers with a hidden mathematical pattern. We need to find the rule governing the transformation from the first number to the second, the second to the third, and the third to the fourth within each set. The rule applies to whole numbers, not their digits.

Set Analysis

Set 1: \(3 – 9 – 81 – 66\)

• From 3 to 9: \(3 \times 3 = 9\) or \(3^2 = 9\).
• From 9 to 81: \(9 \times 9 = 81\) or \(9^2 = 81\).
• From 81 to 66: This doesn't fit the squaring pattern (\(81^2 = 6561\)). Let's explore subtraction. \(81 - 15 = 66\).

Set 2: \(5 – 25 – 625 – 610\)

• From 5 to 25: \(5 \times 5 = 25\) or \(5^2 = 25\).
• From 25 to 625: \(25 \times 25 = 625\) or \(25^2 = 625\).
• From 625 to 610: \(625 - 15 = 610\).

Both sets consistently follow the pattern:

1. Square the current number (\(n \rightarrow n^2\)).
2. Square the result (\((n^2) \rightarrow (n^2)^2\)).
3. Subtract 15 from the latest result (\((n^2)^2 \rightarrow (n^2)^2 - 15\)).

Evaluating Options

Let's apply the derived pattern (\(n \rightarrow n^2 \rightarrow (n^2)^2 \rightarrow (n^2)^2 - 15\)) to the given options.

Option 1: \(2 – 4 – 16 – 21\)

• \(2^2 = 4\) (Correct)
• \(4^2 = 16\) (Correct)
• \(16 - 15 = 1\) (Incorrect, expected 21)

Option 2: \(9 – 81 – 6588 – 6573\)

• \(9^2 = 81\) (Correct)
• \(81^2 = 6561\) (Incorrect, expected 6588)

Option 3: \(4 – 16 – 256 – 241\)

• \(4^2 = 16\) (Correct)
• \(16^2 = 256\) (Correct)
• \(256 - 15 = 241\) (Correct)

This option follows the established pattern.

Option 4: \(7 – 49 – 2150 – 2250\)

• \(7^2 = 49\) (Correct)
• \(49^2 = 2401\) (Incorrect, expected 2150)

The set of operations identified applies correctly only to Option 3.