The next number of the sequence 2 , 6 , 12 , 20 , 30 , 42 , 56 , _____ is

Options

A. 60
B. 64
C. 68
D. 72

Answer

To find the next number in the sequence 2, 6, 12, 20, 30, 42, 56, we will analyze the pattern in the sequence step by step.

### Step 1: Identify the sequence
The given sequence is:
2, 6, 12, 20, 30, 42, 56

### Step 2: Calculate the differences between consecutive terms
Let's find the difference between each pair of consecutive terms:
- 6 - 2 = 4
- 12 - 6 = 6
- 20 - 12 = 8
- 30 - 20 = 10
- 42 - 30 = 12
- 56 - 42 = 14

So, the differences are:
4, 6, 8, 10, 12, 14

### Step 3: Identify the pattern in the differences
The differences themselves form a sequence:
4, 6, 8, 10, 12, 14

We can see that the differences are increasing by 2 each time.

### Step 4: Predict the next difference
Following the pattern, the next difference after 14 would be:
14 + 2 = 16

### Step 5: Add the next difference to the last term of the original sequence
Now, we add this next difference to the last term of the original sequence:
56 + 16 = 72

### Conclusion
Thus, the next number in the sequence is **72**.