Question

Identify the missing number: 2, 6, 12, 20, 30, 42, 56, (___)

• A. 60
• B. 64
• C. 72
• D. 70

Hint:

Formula for sequence $2, 6, 12, 20, 30, 42, 56, \dots$ is $T_n = n(n+1) = n^2 + n$.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Analyzing the mathematical pattern and successive differences in the given numerical sequence.

Step 2: Key Formula or Approach:
The differences between consecutive terms are:
\[6 - 2 = 4,\quad 12 - 6 = 6,\quad 20 - 12 = 8,\quad 30 - 20 = 10,\quad 42 - 30 = 12,\quad 56 - 42 = 14\] The difference increases by 2 at each step ($+4, +6, +8, +10, +12, +14, +16$).
Alternatively, each term follows the pattern $n(n+1)$ for $n = 1, 2, 3, 4, 5, 6, 7, 8$:
\[1\times 2 = 2,\; 2\times 3 = 6,\; 3\times 4 = 12,\; 4\times 5 = 20,\; 5\times 6 = 30,\; 6\times 7 = 42,\; 7\times 8 = 56,\; 8\times 9 = 72\]

Step 3: Detailed Explanation:
Adding 16 to the last term:
\[56 + 16 = 72\]

Step 4: Final Answer:
Therefore, the missing number is 72.