Step 1: Understanding the Question:
This is a number series problem where we need to identify the mathematical rule governing the progression. Step 2: Key Formula or Approach:
Calculate the difference between consecutive terms to see if a secondary pattern (arithmetic progression) exists. Step 3: Detailed Explanation:
Term 1 to Term 2: \(7 - 4 = 3\)
Term 2 to Term 3: \(12 - 7 = 5\)
Term 3 to Term 4: \(19 - 12 = 7\)
Term 4 to Term 5: \(28 - 19 = 9\)
Analyzing the Differences: The sequence of differences is 3, 5, 7, 9...
Pattern Discovery: These are consecutive odd numbers. The next difference should be 11.
Calculation: Next Term = \(28 + 11 = 39\).
Alternative Logic: The series follows the pattern \(n^2 + 3\):
- \(1^2 + 3 = 4\)
- \(2^2 + 3 = 7\)
- \(3^2 + 3 = 12\)
- \(4^2 + 3 = 19\)
- \(5^2 + 3 = 28\)
- \(6^2 + 3 = 36 + 3 = 39\). Step 4: Final Answer:
The missing number in the series is 39. Option (C) is correct.