4, 6, 34, ?, 504, 1234

Options

A. 194
B. 160
C. 186
D. 156

Explanation

To solve the series: 4, 6, 34, ?, 504, 1234, we will analyze the pattern in the differences between the numbers.

### Step-by-Step Solution:

1. **Identify the given series**:
The series is 4, 6, 34, ?, 504, 1234.

2. **Calculate the differences between consecutive terms**:
- Difference between 6 and 4:
\( 6 - 4 = 2 \)
- Difference between 34 and 6:
\( 34 - 6 = 28 \)
- Let the unknown term be \( x \).
- Difference between \( x \) and 34:
\( x - 34 \)
- Difference between 504 and \( x \):
\( 504 - x \)

3. **Set up the differences**:
The differences we have so far are:
- \( 2 \) (from 4 to 6)
- \( 28 \) (from 6 to 34)
- \( x - 34 \) (from 34 to \( x \))
- \( 504 - x \) (from \( x \) to 504)

4. **Look for a pattern in the differences**:
The differences are increasing significantly. Let's analyze the pattern:
- The first difference is \( 2 \).
- The second difference is \( 28 \).
- The next difference should follow a pattern.

5. **Recognize a potential pattern**:
The differences seem to be related to cubes of odd numbers plus one:
- \( 1^3 + 1 = 2 \)
- \( 3^3 + 1 = 28 \)
- The next term should be \( 5^3 + 1 \).

6. **Calculate the next difference**:
- \( 5^3 = 125 \)
- Therefore, the next difference is \( 125 + 1 = 126 \).

7. **Find the unknown term**:
- Now, we can find \( x \):
- \( x - 34 = 126 \)
- So, \( x = 34 + 126 = 160 \).

8. **Verify the next difference**:
- The difference between 504 and 160 should also follow the pattern:
- Next difference should be \( 7^3 + 1 \):
- \( 7^3 = 343 \)
- Therefore, the next difference is \( 343 + 1 = 344 \).
- Check: \( 504 - 160 = 344 \).

9. **Conclusion**:
The missing term in the series is \( 160 \).

### Final Answer:
The missing term is **160**.