In each of the following questions one number is missing to understand the pattern of the series and then insert the missing number .
2,6,12,20, ....... ,42,56,72,90

Options

A. 20
B. 21
C. 30
D. 12

Answer

To solve the problem of finding the missing number in the series 2, 6, 12, 20, ..., 42, 56, 72, 90, we will follow these steps:

### Step 1: Identify the Pattern
First, we need to observe the differences between consecutive numbers in the series.

- From 2 to 6: \(6 - 2 = 4\)
- From 6 to 12: \(12 - 6 = 6\)
- From 12 to 20: \(20 - 12 = 8\)

So far, we have the differences: 4, 6, and 8.

### Step 2: Continue the Pattern
Next, we can see that the differences are increasing by 2 each time. Let's continue this pattern:

- The next difference after 8 should be \(8 + 2 = 10\).
- The difference after that would be \(10 + 2 = 12\).
- Following that, the next difference would be \(12 + 2 = 14\).
- Finally, the difference after that would be \(14 + 2 = 16\).

### Step 3: Calculate the Missing Number
Now, we can apply these differences to find the missing number in the series.

- The last known number before the missing number is 20.
- We will add the next difference (10) to 20 to find the missing number:

\[ 20 + 10 = 30 \]

### Step 4: Verify the Subsequent Numbers
Now, let's verify if this fits with the rest of the series:

- After 30, we should add the next difference (12):
\[ 30 + 12 = 42 \]
- Then, add the next difference (14):
\[ 42 + 14 = 56 \]
- Next, add the next difference (16):
\[ 56 + 16 = 72 \]
- Finally, add the next difference (18):
\[ 72 + 18 = 90 \]

### Conclusion
The missing number in the series is **30**.