Analyzing the Number Pattern

The given sequence is $48 64 80 24 ? 40 12 16 20$. By carefully observing the sequence, we can identify three interleaved arithmetic progressions.

Identifying the Progressions

• Progression 1: Starts with $48$. The numbers are $48, 64, 80$. The common difference is \( 64 - 48 = 16 \) and \( 80 - 64 = 16 \).
• Progression 2: This progression contains the missing number. The known numbers are $24$ and $40$, positioned around the question mark: $24, ?, 40$.
• Progression 3: Starts with $12$. The numbers are $12, 16, 20$. The common difference is \( 16 - 12 = 4 \) and \( 20 - 16 = 4 \).

Calculating the Missing Number

The structure of the sequence suggests that the second progression ($24, ?, 40$) also follows an arithmetic pattern. Let the missing number be \( x \). We assume a constant common difference within this progression.

Therefore, the difference between the second and first term must equal the difference between the third and second term:

$ x - 24 = 40 - x $

To solve for \( x \):

1. Add \( x \) to both sides: \( 2x - 24 = 40 \)
2. Add $24$ to both sides: \( 2x = 40 + 24 \)
3. Simplify: \( 2x = 64 \)
4. Divide by $2$: \( x = \frac{64}{2} \)
5. Calculate the result: \( x = 32 \)

The common difference for this progression is \( 32 - 24 = 8 \) and \( 40 - 32 = 8 \).

Conclusion

The number that correctly replaces the question mark is $32$. The sequence is formed by interleaving three arithmetic progressions with common differences of $16$, $8$, and $4$ respectively.