Complete the number series by choosing the missing terms from the options.
32, 64,256, 1536, ?,122880, 1474560

Options

A. 122860
B. 12268
C. 12285
D. 12288

Answer

To solve the number series: 32, 64, 256, 1536, ?, 122880, 1474560, we need to identify the pattern in the series.

### Step-by-Step Solution:

1. **Identify the Pattern**:
- Let's write down the series: 32, 64, 256, 1536, ?, 122880, 1474560.
- We will look for a relationship between consecutive terms.

2. **Calculate the Ratios**:
- \( \frac{64}{32} = 2 \)
- \( \frac{256}{64} = 4 \)
- \( \frac{1536}{256} = 6 \)
- Notice that the ratios are increasing: 2, 4, 6.

3. **Determine the Next Ratio**:
- Following the pattern, the next ratio should be 8 (continuing the sequence of even numbers: 2, 4, 6, 8).

4. **Calculate the Missing Term**:
- To find the missing term (let's call it \( x \)), we multiply the last known term (1536) by the next ratio (8):
\[ x = 1536 \times 8 = 12288 \]

5. **Verify the Next Term**:
- Now, we check if this term fits with the next term in the series:
- \( \frac{122880}{12288} = 10 \) (which is consistent with the increasing pattern of ratios).

6. **Final Series**:
- The complete series is: 32, 64, 256, 1536, 12288, 122880, 1474560.

### Conclusion:
The missing term in the series is **12288**, which corresponds to option D.