Direction: What should come in place of question mark in the following number series?
81 91 109 146 221 351 ?

Options

A. 578
B. 582
C. 591
D. 576

Answer

To find the missing number in the series: 81, 91, 109, 146, 221, 351, ?, we will analyze the pattern of the differences between consecutive numbers.

### Step-by-Step Solution:

1. **Identify the series**:
The series is: 81, 91, 109, 146, 221, 351.

2. **Calculate the differences**:
- 91 - 81 = 10
- 109 - 91 = 18
- 146 - 109 = 37
- 221 - 146 = 75
- 351 - 221 = 130

So, the differences are: 10, 18, 37, 75, 130.

3. **Calculate the differences of the differences**:
- 18 - 10 = 8
- 37 - 18 = 19
- 75 - 37 = 38
- 130 - 75 = 55

So, the second differences are: 8, 19, 38, 55.

4. **Calculate the differences of the second differences**:
- 19 - 8 = 11
- 38 - 19 = 19
- 55 - 38 = 17

So, the third differences are: 11, 19, 17.

5. **Identify the pattern in the third differences**:
The third differences do not show a clear pattern, so we will look at the second differences again.

6. **Calculate the next second difference**:
The second differences seem to be increasing. If we assume the next difference follows the pattern, we can estimate it. The last second difference was 55. If we add the next increment (which we can estimate as around 19), we get:
- Next second difference = 55 + 19 = 74.

7. **Calculate the next first difference**:
Now, we will add this new second difference to the last first difference:
- Next first difference = 130 + 74 = 204.

8. **Calculate the next number in the series**:
Finally, we will add this new first difference to the last number in the series:
- Next number = 351 + 204 = 555.

Thus, the missing number in the series is **555**.