Step 1: Understanding the Concept:
We need to identify the underlying pattern in the given numerical sequence to determine the missing term. Step 2: Key Formula or Approach:
There are two common ways to approach this:
1. Method of Differences: Calculate the differences between consecutive terms and see if a pattern emerges.
2. Formula Method: Try to find a formula for the n-th term of the sequence. The numbers are products of consecutive integers. Step 3: Detailed Explanation: Method 1: Analyzing the Differences
Let's find the difference between consecutive terms:
- \(6 - 2 = 4\)
- \(12 - 6 = 6\)
- \(20 - 12 = 8\)
The differences are 4, 6, 8. This is an arithmetic progression with a common difference of 2.
The next difference in the sequence should be \(8 + 2 = 10\).
So, the missing term is \(20 + 10 = 30\).
Let's check if the next term fits the pattern. The next difference would be \(10 + 2 = 12\).
\(30 + 12 = 42\), which is the last term given. The pattern is consistent.
Method 2: Finding a General Formula
Let's examine the terms themselves:
- \(2 = 1 \times 2\)
- \(6 = 2 \times 3\)
- \(12 = 3 \times 4\)
- \(20 = 4 \times 5\)
The pattern for the n-th term \(a_n\) is \(a_n = n \times (n+1)\).
The missing term is the 5th term in the sequence (\(a_5\)).
\[ a_5 = 5 \times (5+1) = 5 \times 6 = 30 \]
The next term is \(a_6\):
\[ a_6 = 6 \times (6+1) = 6 \times 7 = 42 \], which matches the sequence. Step 4: Final Answer:
Both methods confirm that the missing number in the sequence is 30. This corresponds to option (C).