Select the number that logically completes the following numerical sequence.

24, 120, 720, 5040, 40320, _____

The correct answer is 362880 (which corresponds to Option 3: 362880).

Step-by-Step Explanation:

To find the missing number in the given sequence, let us analyze the mathematical relationship between consecutive terms.

Given sequence: 24, 120, 720, 5040, 40320, _____

Let us look at the pattern of multiplication between each term:

1st term to 2nd term:

$24\times 5=120$

2nd term to 3rd term:

$120\times 6=720$

3rd term to 4th term:

$720\times 7=5040$

4th term to 5th term:

$5040\times 8=40320$

We can observe a clear pattern where each successive term is obtained by multiplying the preceding term by consecutive integers starting from 5 (i.e., × 5, × 6, × 7, × 8, ...).

Following this logical pattern, the next term must be obtained by multiplying the 5th term by 9:

$40320\times 9=362880$

Thus, the number that logically completes the sequence is 362880.