Step 1: Understanding the Concept:
This series represents a quadratic growth pattern or a sequence of rectangular numbers. We examine the differences between the numbers to find a common increment.
Step 2: Key Formula or Approach:
Calculate the first-level differences: \( d_1 = T_2 - T_1 \), \( d_2 = T_3 - T_2 \), and so on.
Step 3: Detailed Explanation:
1. Calculating the Differences:
\[ 6 - 2 = 4 \]
\[ 12 - 6 = 6 \]
\[ 20 - 12 = 8 \]
\[ 30 - 20 = 10 \]
The differences between the terms are 4, 6, 8, 10.
2. Identifying the Pattern:
The differences themselves form an arithmetic progression of consecutive even numbers starting from 4.
The next difference in the sequence should be: \( 10 + 2 = 12 \).
3. Finding the Missing Term:
To find the next term, we add the next difference (12) to the last known term (30).
\[ 30 + 12 = 42 \]
4. Alternative Logic (Square Pattern):
This series follows the rule \( n^2 + n \):
\( 1^2 + 1 = 2 \)
\( 2^2 + 2 = 6 \)
\( 3^2 + 3 = 12 \)
\( 4^2 + 4 = 20 \)
\( 5^2 + 5 = 30 \)
The next term is \( 6^2 + 6 = 36 + 6 = 42 \).
5. Alternative Logic (Product of Consecutives):
\( 1 \times 2 = 2 \)
\( 2 \times 3 = 6 \)
\( 3 \times 4 = 12 \)
\( 4 \times 5 = 20 \)
\( 5 \times 6 = 30 \)
The next term is \( 6 \times 7 = 42 \).
Step 4: Final Answer:
All logical pathways confirm the missing term is 42.