2 14 84 420 1680 5040 ?

Options

A. 8940
B. 8680
C. 10080
D. 5030

Explanation

To solve the series: 2, 14, 84, 420, 1680, 5040, ?, we need to identify the pattern in the series.

### Step-by-Step Solution:

1. **Identify the Pattern**:
The series appears to be increasing rapidly. To find a pattern, we can look at the ratios of consecutive terms.

2. **Calculate Ratios**:
- Divide the second term by the first term:
\[ \frac{14}{2} = 7 \]
- Divide the third term by the second term:
\[ \frac{84}{14} = 6 \]
- Divide the fourth term by the third term:
\[ \frac{420}{84} = 5 \]
- Divide the fifth term by the fourth term:
\[ \frac{1680}{420} = 4 \]
- Divide the sixth term by the fifth term:
\[ \frac{5040}{1680} = 3 \]

3. **Observe the Pattern in Ratios**:
The ratios we calculated are: 7, 6, 5, 4, 3. We can see that the ratios are decreasing by 1 each time.

4. **Predict the Next Ratio**:
Following the established pattern, the next ratio should be:
\[ 2 \quad (\text{since } 3 - 1 = 2) \]

5. **Calculate the Next Term**:
To find the next term in the series, we multiply the last term (5040) by the next ratio (2):
\[ 5040 \times 2 = 10080 \]

6. **Conclusion**:
Therefore, the next number in the series is **10080**.

### Final Answer:
The next number in the series is **10080**.

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