Find the missing term in the given series?
29, 31, 34, 39, ?, 57

Options

A. 48
B. 46
C. 50
D. 51

Answer

To find the missing term in the series 29, 31, 34, 39, ?, 57, we can analyze the differences between the consecutive terms.

1. **Identify the series**:
The given series is:
\[ 29, 31, 34, 39, ?, 57 \]

2. **Calculate the differences between consecutive terms**:
- The difference between 29 and 31 is:
\[ 31 - 29 = 2 \]
- The difference between 31 and 34 is:
\[ 34 - 31 = 3 \]
- The difference between 34 and 39 is:
\[ 39 - 34 = 5 \]
- The difference between 39 and the missing term (let's call it \( x \)) is unknown.
- The difference between the missing term \( x \) and 57 is:
\[ 57 - x \]

3. **List the differences**:
So far, we have:
\[ 2, 3, 5, ?, (57 - x) \]

4. **Identify a pattern in the differences**:
The differences we have are:
- 2 (which is the 1st prime number)
- 3 (which is the 2nd prime number)
- 5 (which is the 3rd prime number)

The next prime number after 5 is 7.

5. **Calculate the missing term using the identified pattern**:
If we add 7 to 39:
\[ 39 + 7 = 46 \]
So, the missing term is 46.

6. **Verify the next difference**:
Now, we check the difference between 46 and 57:
\[ 57 - 46 = 11 \]
Here, 11 is also a prime number, which fits our pattern.

7. **Conclusion**:
Thus, the missing term in the series is:
\[ \boxed{46} \]