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In mathematics, a factorial is an important concept used to calculate permutations, combinations, and sequence problems. A factorial is denoted by an exclamation mark after a number, like this: \( n! \). This represents the product of all positive integers up to \( n \).
For example, if we look at \( 3! \), it equals \( 3 \times 2 \times 1 = 6 \). Here are some key points to remember about factorials:

• \( 0! = 1 \) by definition.
• The factorial function grows very rapidly as \( n \) increases.
• Factorials are only defined for non-negative integers.

Factorials play a crucial role in calculating terms in sequences where each term involves a factorial, like in our given sequence, \( a_n = \frac{n!}{n^2 - n - 1} \). Understanding how to compute factorials and their properties is essential for solving such sequence problems efficiently.

A rational expression is a fraction where both the numerator and the denominator are polynomials. In our sequence formula, \( a_n = \frac{n!}{n^2 - n - 1} \), the numerator is the factorial \( n! \), and the denominator is the polynomial \( n^2 - n - 1 \).
Here are some important points about rational expressions:

• They can sometimes be simplified by factoring and reducing common factors.
• The expression is undefined whenever the denominator equals zero. So, it’s important to identify values of \( n \) that make the equation undefined, which happens when \( n^2 - n - 1 = 0 \).
• Rational expressions provide a method to represent sequences and functions, helping to understand how they behave.

Recognizing how to handle and simplify rational expressions is a vital skill, especially when dealing with complex sequence calculations.

Calculating sequences involves finding successive terms based on a defined formula. A sequence is an ordered list of numbers, and each number is called a term. The formula you've given allows us to find the first four terms by plugging different positive integers into the sequence function.
Key considerations when calculating sequences include:

• Understand what kind of sequence you are dealing with—arithmetic, geometric, or another type like the one given.
• Substitute values for \( n \) systematically to find each term in a sequence up to the point you're interested in (e.g., the first four terms).
• Each term is calculated independently using the sequence formula, even though patterns might emerge as you calculate more terms.

For the sequence \( a_n = \frac{n!}{n^2 - n - 1} \), you calculate each term by computing \( n! \), the polynomial in the denominator, and then dividing, following through with simplification if possible. This process illustrates how individual sequence terms are generated from a given formula.