Finding the Missing Number in the Series

Let's analyze the given number series: 4, 12, 48, 240, 1440, ? We need to identify the pattern that connects the numbers in this sequence to find the missing number.

Analyzing the Number Pattern

We can look at the relationship between consecutive terms in the series. Let's see how each term is obtained from the previous term:

• From 4 to 12: We can observe that \(4 \times 3 = 12\).
• From 12 to 48: We can observe that \(12 \times 4 = 48\).
• From 48 to 240: We can observe that \(48 \times 5 = 240\).
• From 240 to 1440: We can observe that \(240 \times 6 = 1440\).

Identifying the Pattern Rule

From the analysis above, it is clear that the pattern involves multiplying the previous term by a consecutive integer. The multiplier starts from 3 and increases by 1 for each subsequent step:

• First term is 4.
• Second term = First term \( \times 3 = 4 \times 3 = 12\).
• Third term = Second term \( \times 4 = 12 \times 4 = 48\).
• Fourth term = Third term \( \times 5 = 48 \times 5 = 240\).
• Fifth term = Fourth term \( \times 6 = 240 \times 6 = 1440\).

Calculating the Missing Term

Following this established number pattern, the next multiplier in the series should be 7. Therefore, to find the missing number, we need to multiply the last given term (1440) by 7.

Missing term = Fifth term \( \times 7 = 1440 \times 7 \)

Let's calculate the value:

\(1440 \times 7 = 10080\)

Thus, the missing number in the series 4, 12, 48, 240, 1440, ? is 10080.

Conclusion

The missing number in the given number series is 10080, following the pattern of multiplying by consecutive integers starting from 3.