What is the 8th term of the sequence 1, 2, 6, 24, 120? This sequence is the factorial sequence, where each term is the product of all positive integers up to a certain number. The 8th term of this sequence is 40,320.
Understanding the Factorial Sequence
The sequence 1, 2, 6, 24, 120 corresponds to the factorials of numbers starting from 1. The factorial of a number ( n ), denoted as ( n! ), is the product of all positive integers less than or equal to ( n ).
How is the Factorial Sequence Formed?
- 1! = 1: The factorial of 1 is 1.
- 2! = 2 \times 1 = 2: The factorial of 2 is 2.
- 3! = 3 \times 2 \times 1 = 6: The factorial of 3 is 6.
- 4! = 4 \times 3 \times 2 \times 1 = 24: The factorial of 4 is 24.
- 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120: The factorial of 5 is 120.
This pattern continues, with each term being the product of the number and all positive integers before it.
Calculating the 8th Term
To find the 8th term of the sequence, calculate ( 8! ):
- 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720
- 7! = 7 \times 6! = 7 \times 720 = 5,040
- 8! = 8 \times 7! = 8 \times 5,040 = 40,320
Thus, the 8th term is 40,320.
Why is the Factorial Sequence Important?
Applications in Mathematics and Science
Factorials are used in various mathematical and scientific applications, including:
- Combinatorics: Calculating permutations and combinations.
- Probability Theory: Determining probabilities in complex scenarios.
- Algebra and Calculus: Solving series and limits.
Practical Examples and Case Studies
- Permutations: If you have 8 books and want to know how many ways you can arrange them on a shelf, you calculate ( 8! = 40,320 ).
- Combinations: When choosing 3 books from 8, factorials help calculate the number of possible selections.
Comparison of Factorial Terms
Here’s a table comparing the first few terms of the factorial sequence:
| Term Number | Factorial Expression | Value |
|---|---|---|
| 1 | 1! | 1 |
| 2 | 2! | 2 |
| 3 | 3! | 6 |
| 4 | 4! | 24 |
| 5 | 5! | 120 |
| 6 | 6! | 720 |
| 7 | 7! | 5,040 |
| 8 | 8! | 40,320 |
People Also Ask
What is a factorial used for?
Factorials are used in mathematics for permutations and combinations, which are essential in probability and statistics. They also appear in algebra and calculus when dealing with series and limits.
How do you calculate a factorial?
To calculate a factorial, multiply the number by every positive integer less than it. For example, ( 5! ) is calculated as ( 5 \times 4 \times 3 \times 2 \times 1 = 120 ).
Why is 0! equal to 1?
The value of 0! is defined as 1 to maintain the consistency of the factorial function in combinatorics, particularly when calculating the number of ways to arrange zero objects.
Can factorials be negative?
Factorials are defined only for non-negative integers. The concept of factorials does not extend to negative numbers in standard mathematics.
How are factorials used in real life?
Factorials are used in fields like engineering and computer science for solving problems involving permutations, combinations, and probability distributions.
Conclusion
The 8th term of the sequence 1, 2, 6, 24, 120 is 40,320, calculated as ( 8! ). Understanding factorials is crucial for various mathematical applications, from solving complex probability problems to computing permutations and combinations. For further exploration, consider reading about how factorials are applied in different scientific fields, such as computer algorithms and statistical analysis.
