This question asks us to find the missing number in the given sequence: 1, 2, 6, ______, 120, 720.

To solve this type of problem, we need to look for a pattern or rule that connects the numbers in the sequence.

Analyzing the Missing Number Sequence

Let's examine the relationship between consecutive terms in the sequence:

• From the first term (1) to the second term (2): The difference is $2 - 1 = 1$. The ratio is $2 / 1 = 2$.
• From the second term (2) to the third term (6): The difference is $6 - 2 = 4$. The ratio is $6 / 2 = 3$.

Looking at the ratios, we see they are 2 and 3. This suggests a possible pattern involving multiplication by increasing integers.

Identifying the Pattern in the Sequence

Let's hypothesize that each term is obtained by multiplying the previous term by a consecutive integer, starting from 2.

• Term 1: 1
• Term 2: $1 \times 2 = 2$ (Matches the given sequence)
• Term 3: $2 \times 3 = 6$ (Matches the given sequence)
• Term 4: If the pattern continues, the next multiplier should be 4. So, the missing term would be $6 \times 4$.

Calculating the Missing Term

Based on the identified pattern (multiplying by 2, then 3, then 4, and so on), the missing term is calculated as:

Missing Term = Term 3 $\times$ 4

Missing Term = $6 \times 4 = 24$

Verifying the Pattern with Subsequent Terms

Let's check if the pattern holds for the terms after the missing number (which we predict is 24).

• The next multiplier after 4 should be 5. So, Term 5 should be $24 \times 5$.
• Term 5: $24 \times 5 = 120$ (Matches the given sequence)

Now, the next multiplier after 5 should be 6. So, Term 6 should be $120 \times 6$.

• Term 6: $120 \times 6 = 720$ (Matches the given sequence)

The pattern of multiplying by consecutive integers (2, 3, 4, 5, 6) perfectly matches the given sequence. Therefore, the missing number is indeed 24.

Understanding Mathematical Sequence Patterns

Sequences can follow various patterns. Some common types include:

• Arithmetic Sequence: A constant difference between consecutive terms (e.g., 2, 4, 6, 8...).
• Geometric Sequence: A constant ratio between consecutive terms (e.g., 3, 6, 12, 24...).
• Fibonacci Sequence: Each term is the sum of the two preceding terms (e.g., 1, 1, 2, 3, 5...).
• Patterns involving squares, cubes, or other powers.
• Patterns involving increasing/decreasing differences or ratios (like the sequence in this question).

Identifying the pattern is the key to solving sequence problems.

Revision Table: Sequence Pattern Types

Additional Information: Recognizing Number Patterns

Here are some tips for identifying patterns in number sequences:

• Look at the difference between consecutive terms. Is it constant? Is it changing in a predictable way (e.g., increasing by a constant)?
• Look at the ratio between consecutive terms. Is it constant? Is it changing predictably?
• Consider operations other than addition/subtraction or multiplication/division, such as squares, cubes, or factorials.
• Look for patterns that involve the position of the term in the sequence (e.g., $n^2$, $2n-1$).
• Sometimes the pattern involves alternating operations or combining different simple patterns.

Practice with various types of sequences helps improve pattern recognition skills.