Understanding the Number Series Pattern

The question asks us to find the missing number in the given series: 1, 2, 6, 24, 120, ___.

To solve this, we need to identify the pattern or rule that connects consecutive numbers in the series.

Analyzing the Pattern in the Series

Let's look at the relationship between each pair of consecutive numbers:

• From the 1st term to the 2nd term: 1 to 2. We can see that $1 \times 2 = 2$.
• From the 2nd term to the 3rd term: 2 to 6. We can see that $2 \times 3 = 6$.
• From the 3rd term to the 4th term: 6 to 24. We can see that $6 \times 4 = 24$.
• From the 4th term to the 5th term: 24 to 120. We can see that $24 \times 5 = 120$.

We observe a clear pattern here. Each term is obtained by multiplying the previous term by a consecutive integer, starting from 2.

• Term 1: 1
• Term 2: Term 1 $\times 2 = 1 \times 2 = 2$
• Term 3: Term 2 $\times 3 = 2 \times 3 = 6$
• Term 4: Term 3 $\times 4 = 6 \times 4 = 24$
• Term 5: Term 4 $\times 5 = 24 \times 5 = 120$

Calculating the Missing Number

Following this established pattern, the next multiplier for the 6th term should be 6 (the next consecutive integer after 5).

Therefore, the missing number (the 6th term) is:

Term 6 = Term 5 $\times 6$

Term 6 = $120 \times 6$

Let's perform the multiplication:

$120 \times 6 = 720$

So, the missing number in the series is 720.

Conclusion

The series follows the pattern where each term is the product of the previous term and an increasing sequence of integers starting from 2. By applying this pattern, the next number after 120 is 720.