Directions: Insert the missing number: 2, 6, 12, 20, 30, 42, 56, (.....)

### Concept & Logic The sequence can be understood through the product of consecutive integers or by analyzing the progressively increasing differences between terms. ### Step-by-Step Solution **Method 1: Product of Consecutive Integers** Observe the terms: - $1 \times 2 = 2$ - $2 \times 3 = 6$ - $3 \times 4 = 12$ - $4 \times 5 = 20$ - $5 \times 6 = 30$ - $6 \times 7 = 42$ - $7 \times 8 = 56$ The next term must be the product of 8 and 9. $8 \times 9 = 72$. **Method 2: Step-based Differences** Let's find the differences between terms: - $6 - 2 = 4$ - $12 - 6 = 6$ - $20 - 12 = 8$ - $30 - 20 = 10$ - $42 - 30 = 12$ - $56 - 42 = 14$ The differences form an arithmetic progression (4, 6, 8, 10, 12, 14) increasing by 2. The next difference will be $14 + 2 = 16$. So, $56 + 16 = 72$. ### Exam Strategy & Shortcut Familiarize yourself with the $n(n+1)$ sequence ($2, 6, 12, 20, 30...$) as it frequently appears as the core logic in many aptitude tests. ### Common Pitfall Calculating the correct difference sequence but forgetting to add the final difference to the last term, simply identifying the difference pattern but executing it improperly. ### Final Answer Therefore, the correct answer is **72**.