15 ? 24 33 97 122

Options

A. 20
B. 19
C. 17
D. 16

Explanation

To find the missing number in the series: 15, ?, 24, 33, 97, 122, we will analyze the differences between the numbers and identify a pattern.

### Step-by-Step Solution:

1. **Identify the Series**: The given series is 15, ?, 24, 33, 97, 122.

2. **Calculate Differences**: We will calculate the differences between consecutive numbers in the series:
- Difference between 24 and 33:
\[ 33 - 24 = 9 \]
- Difference between 33 and 97:
\[ 97 - 33 = 64 \]
- Difference between 97 and 122:
\[ 122 - 97 = 25 \]

3. **List the Differences**: Now we have the differences:
- \( ? - 15 \)
- \( 9 \) (from 24 to 33)
- \( 64 \) (from 33 to 97)
- \( 25 \) (from 97 to 122)

4. **Identify the Pattern**: We can see that:
- \( 9 = 3^2 \)
- \( 64 = 4^3 \)
- \( 25 = 5^2 \)

The pattern seems to alternate between squares and cubes of consecutive integers:
- The first difference should be \( 2^3 = 8 \).

5. **Calculate the Missing Number**:
- If the first difference is \( 8 \), then:
\[ ? = 15 + 8 = 23 \]

6. **Verify the Pattern**: Now we will check if this fits in the series:
- The differences would now be:
- \( 23 - 15 = 8 \)
- \( 24 - 23 = 1 \)
- \( 33 - 24 = 9 \)
- \( 97 - 33 = 64 \)
- \( 122 - 97 = 25 \)

This confirms that the pattern holds.

7. **Conclusion**: The missing number in the series is:
\[ \text{The missing number is } 23. \]