Number Sequence Pattern Identification

The question asks us to find the missing number in the sequence: 1, 2, 6, 24, 120, ____. We need to identify the pattern governing this sequence to determine the next term.

Analyzing the Sequence Terms

Let's look at the given numbers:

• Term 1: 1
• Term 2: 2
• Term 3: 6
• Term 4: 24
• Term 5: 120

Finding the Pattern Rule

We can examine the relationship between consecutive terms:

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

The pattern observed is that each term is generated by multiplying the previous term by an integer, and this integer increases by 1 for each subsequent step (multiplying by 2, then 3, then 4, then 5).

This sequence is also recognizable as the sequence of factorials:

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

Calculating the Missing Term

Following the identified pattern, the next multiplier should be 6.

We multiply the last term (120) by 6:

Calculation: $120 \times 6 = 720$

Alternatively, using the factorial pattern, the next term is $6!$:

Calculation: $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$

Therefore, the missing number in the sequence is 720.