Understanding the Number Series

The question requires us to find the missing number in a given numerical sequence. We need to identify the underlying pattern or rule governing the series and apply it to determine the value that should replace the question mark (?). The series provided is: 8, 4, 4, 6, 12, 30, ?

Analyzing the Series Pattern

To find the missing number, let's examine the relationship between each consecutive pair of numbers in the series:

• The first term is 8, and the second is 4. The transition can be described as multiplication by 0.5 (or division by 2): $$8 \times 0.5 = 4$$
• The second term is 4, and the third is 4. This transition is multiplication by 1: $$4 \times 1 = 4$$
• The third term is 4, and the fourth is 6. This transition is multiplication by 1.5: $$4 \times 1.5 = 6$$
• The fourth term is 6, and the fifth is 12. This transition is multiplication by 2: $$6 \times 2 = 12$$
• The fifth term is 12, and the sixth is 30. This transition is multiplication by 2.5: $$12 \times 2.5 = 30$$

Identifying the Multiplier Sequence

By analyzing these transitions, we can observe a sequence of multipliers: 0.5, 1, 1.5, 2, 2.5. This sequence represents an arithmetic progression, where each term increases by a constant difference of 0.5.

Calculating the Next Term Based on the Pattern

To find the next multiplier in the sequence, we add the common difference (0.5) to the last multiplier (2.5):

$$ \text{Next Multiplier} = 2.5 + 0.5 = 3 $$

Now, we apply this multiplier to the last term of the series (30) to find the missing number:

$$ 30 \times 3 = 90 $$

Therefore, according to the identified pattern of an arithmetic progression in the multipliers, the next number in the series should logically be 90.