24, 536, 487, 703, 678, ?

Options

A. 768
B. 748
C. 764
D. 742

Explanation

To solve the problem of finding the missing number in the series: 24, 536, 487, 703, 678, ?, we will analyze the differences between consecutive terms and identify a pattern.

### Step-by-step Solution:

1. **Identify the series**: The given series is 24, 536, 487, 703, 678, ?.

2. **Calculate the differences between consecutive terms**:
- From 24 to 536:
\[ 536 - 24 = 512 \]
- From 536 to 487:
\[ 487 - 536 = -49 \]
- From 487 to 703:
\[ 703 - 487 = 216 \]
- From 703 to 678:
\[ 678 - 703 = -25 \]

So, the differences are:
- 512, -49, 216, -25.

3. **Analyze the pattern of differences**:
- The differences can be expressed as:
- 512 = \(8^3\) (cube of 8)
- -49 = \(-7^2\) (negative square of 7)
- 216 = \(6^3\) (cube of 6)
- -25 = \(-5^2\) (negative square of 5)

This suggests a pattern where we alternate between adding cubes and subtracting squares.

4. **Continue the pattern**:
- Following the established pattern, the next difference should be:
- \(4^3 = 64\) (cube of 4)

5. **Calculate the missing number**:
- The last number in the series is 678. We will add the next difference (64) to it:
\[ 678 + 64 = 742 \]

6. **Conclusion**:
- The missing number in the series is **742**.