Number Series Analysis

The given number series is: 4, 4, 6, 12, __?, 90, 315.

We need to find the pattern governing this series to determine the missing number.

Identifying the Pattern

Let's examine the relationship between consecutive terms:

• Term 1 to Term 2: 4 to 4. The operation is multiplication by 1 ($4 \times 1 = 4$).
• Term 2 to Term 3: 4 to 6. The operation is multiplication by 1.5 ($4 \times 1.5 = 6$).
• Term 3 to Term 4: 6 to 12. The operation is multiplication by 2 ($6 \times 2 = 12$).

The multipliers are increasing sequentially: 1, 1.5, 2. The next multiplier should be 2.5.

Calculating the Missing Number

Applying the pattern to find the missing term:

• Term 4 to Term 5 (__?__): Multiply the previous term (12) by the next multiplier (2.5). $12 \times 2.5 = 30$

So, the missing number is 30.

Verifying the Pattern

Let's check if this pattern continues for the remaining terms:

• Term 5 to Term 6: Multiply the missing number (30) by the next multiplier (3). $30 \times 3 = 90$. This matches the series.
• Term 6 to Term 7: Multiply the previous term (90) by the next multiplier (3.5). $90 \times 3.5 = 315$. This also matches the series.

The pattern holds true. The sequence of multipliers is 1, 1.5, 2, 2.5, 3, 3.5.

Conclusion

The missing number in the series is 30.