What should come in place of question mark (?) in the following number series ?
`"1 6 21 66 201 ?"`

Options

A. 609
B. 603
C. 606
D. 600

Answer

To solve the number series: 1, 6, 21, 66, 201, ?, we need to identify the pattern in the series.

### Step-by-Step Solution:

1. **Identify the Series**: The given series is 1, 6, 21, 66, 201, ?.

2. **Calculate the First Differences**:
- 6 - 1 = 5
- 21 - 6 = 15
- 66 - 21 = 45
- 201 - 66 = 135

So, the first differences are: 5, 15, 45, 135.

3. **Calculate the Second Differences**:
- 15 - 5 = 10
- 45 - 15 = 30
- 135 - 45 = 90

So, the second differences are: 10, 30, 90.

4. **Calculate the Third Differences**:
- 30 - 10 = 20
- 90 - 30 = 60

So, the third differences are: 20, 60.

5. **Calculate the Fourth Differences**:
- 60 - 20 = 40

The fourth difference is: 40.

6. **Identify the Pattern**:
- The first differences are multiplied by 3 each time:
- 5 * 3 = 15
- 15 * 3 = 45
- 45 * 3 = 135

7. **Calculate the Next First Difference**:
- Continuing the pattern, the next first difference would be:
- 135 * 3 = 405.

8. **Calculate the Next Number in the Series**:
- Add the next first difference to the last number in the series:
- 201 + 405 = 606.

9. **Conclusion**: The next number in the series is 606.

### Final Answer:
The number that should come in place of the question mark (?) is **606**.

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