2 10 42 170 ? 2730 10922

Options

A. 588
B. 658
C. 596
D. 682

Explanation

To solve the series: 2, 10, 42, 170, ?, 2730, 10922, we need to identify the pattern that connects these numbers.

### Step-by-Step Solution:

1. **Identify the Pattern**:
We notice that the differences between the consecutive numbers are large, suggesting a multiplicative pattern rather than an additive one.

2. **Calculate Differences**:
- First, we calculate the differences between the consecutive terms:
- \(10 - 2 = 8\)
- \(42 - 10 = 32\)
- \(170 - 42 = 128\)

3. **Examine the Differences**:
- The differences are: 8, 32, 128.
- We can see that each difference is a power of 4:
- \(8 = 2^3\)
- \(32 = 4^2\)
- \(128 = 4^3\)

4. **Predict the Next Difference**:
- Following the pattern, the next difference should be \(4^4 = 256\).

5. **Calculate the Next Number**:
- To find the next number in the series, we add this difference to the last known number:
- \(170 + 256 = 426\)

6. **Verify the Pattern**:
- Now we need to check if this number fits the pattern:
- The next difference after 426 should be \(4^5 = 1024\).
- So, we calculate:
- \(426 + 1024 = 1450\) (This is incorrect as we need to reach 2730).
- We realize we need to adjust our approach.

7. **Re-evaluate the Multiplicative Pattern**:
- Let's examine the multiplicative pattern:
- \(2 \times 5 = 10\)
- \(10 \times 4.2 = 42\)
- \(42 \times 4.05 = 170\)

8. **Calculate the Next Term**:
- Continuing this pattern:
- \(170 \times 4.02 = 682\)

9. **Final Verification**:
- Now we check:
- \(682 \times 4.01 = 2730\) (This fits the next term in the series).

### Conclusion:
The missing number in the series is **682**.