2,6,12,20,30,42,56 (…)

Options

A. 60
B. 64
C. 72
D. 70

Answer

To find the next term in the series 2, 6, 12, 20, 30, 42, 56, we can follow these steps:

### Step 1: Identify the given series
The series provided is:
\[ 2, 6, 12, 20, 30, 42, 56 \]

### Step 2: Calculate the differences between consecutive terms
We will find the differences 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
Next, we notice that the differences themselves form a sequence:
- The differences are increasing by 2 each time:
- \( 6 - 4 = 2 \)
- \( 8 - 6 = 2 \)
- \( 10 - 8 = 2 \)
- \( 12 - 10 = 2 \)
- \( 14 - 12 = 2 \)

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

### Step 5: Calculate the next term in the series
Now, we add this next difference to the last term of the series:
\[ 56 + 16 = 72 \]

### Conclusion
Thus, the next term in the series is:
\[ \text{Next term} = 72 \]