What should come in place of the question mark(?) in the following number series?
24 ? 109 134 150 159

Options

A. 71
B. 65
C. 86
D. None of these

Answer

To find the missing number in the series: 24, ?, 109, 134, 150, 159, we will analyze the differences between the consecutive numbers.

1. **Identify the known numbers in the series:**
- The series is: 24, ?, 109, 134, 150, 159.

2. **Calculate the differences between the known numbers:**
- Difference between 109 and 134:
\[ 134 - 109 = 25 \]
- Difference between 134 and 150:
\[ 150 - 134 = 16 \]
- Difference between 150 and 159:
\[ 159 - 150 = 9 \]

3. **Now, we can calculate the difference between the known numbers backwards:**
- Let’s denote the missing number as \( x \).
- Difference between 24 and \( x \):
\[ x - 24 \]
- Difference between \( x \) and 109:
\[ 109 - x \]

4. **Now, we can set up the equations based on the differences we calculated:**
- We already have the differences as:
- \( 109 - x \) should be equal to \( x - 24 + 25 \) (the difference between 109 and \( x \) should equal the difference between 24 and \( x \) plus 25).
- Rearranging gives:
\[ 109 - x = (x - 24) + 25 \]
- Simplifying:
\[ 109 - x = x + 1 \]
- Bringing all \( x \) terms to one side:
\[ 109 - 1 = 2x \]
\[ 108 = 2x \]
\[ x = 54 \]

5. **Thus, the missing number is:**
\[ \text{The missing number is } 54. \]

### Summary of Steps:
1. Identify the known numbers in the series.
2. Calculate the differences between the known numbers.
3. Set up equations based on the differences.
4. Solve for the missing number.