The task is to find the missing number in the numerical series: 12, 24, 96, 576, ________. We need to identify the underlying pattern and apply it to determine the correct value for the blank space.

Series Pattern Identification

Let's analyze the progression between consecutive numbers in the given sequence:

• The first term is 12.
• The second term is 24. The relationship between the first and second term is found by multiplication: $12 \times 2 = 24$.
• The third term is 96. The relationship between the second and third term is: $24 \times 4 = 96$.
• The fourth term is 576. The relationship between the third and fourth term is: $96 \times 6 = 576$.

By examining these steps, we can see that each term is multiplied by an increasing even number. The multipliers used are 2, 4, and 6.

The pattern of multipliers (2, 4, 6) suggests an arithmetic sequence where the common difference is 2. That is, each multiplier is 2 greater than the previous one ($4 - 2 = 2$, $6 - 4 = 2$).

Therefore, the next multiplier in this pattern should be $6 + 2 = 8$.

Term Calculation

To find the missing term, we apply the identified pattern. We multiply the last known term of the series (576) by the next multiplier in the sequence, which we determined to be 8.

The calculation is:

Missing Term = Last Term $\times$ Next Multiplier

Missing Term = $576 \times 8$

Performing the multiplication:

$576 \times 8 = 4608$

Result

The calculation shows that the missing number required to complete the series is 4608.