Problem 45 - College Algebra

Question

Write the first four terms of the sequence. $$a_{n}=\frac{n !}{n^{2}-n-1}$$

Step-by-Step Solution

Answer

The first four terms are -1, 2, \( \frac{6}{5} \), \( \frac{24}{11} \).

Step 1: Calculate the First Term

For the first term, substitute \( n = 1 \) into the formula. This gives: \[ a_1 = \frac{1!}{1^2 - 1 - 1} = \frac{1}{-1} = -1 \]So, the first term is \( a_1 = -1 \).

Step 2: Calculate the Second Term

Next, substitute \( n = 2 \) into the formula: \[ a_2 = \frac{2!}{2^2 - 2 - 1} = \frac{2}{1} = 2 \]Thus, the second term is \( a_2 = 2 \).

Step 3: Calculate the Third Term

Substitute \( n = 3 \) into the formula: \[ a_3 = \frac{3!}{3^2 - 3 - 1} = \frac{6}{5} \]Therefore, the third term is \( a_3 = \frac{6}{5} \).

Step 4: Calculate the Fourth Term

Finally, substitute \( n = 4 \) into the sequence formula: \[ a_4 = \frac{4!}{4^2 - 4 - 1} = \frac{24}{11} \]Hence, the fourth term is \( a_4 = \frac{24}{11} \).

Key Concepts

Factorial

The factorial of a number is a key concept in mathematics, especially in fields like combinatorics and calculus. It is denoted by an exclamation mark, like this: \( n! \). The factorial of a non-negative integer \( n \) is the product of all positive integers less than or equal to \( n \). Here's the basic idea:

• \( n! = n \times (n - 1) \times (n - 2) \times ... \times 2 \times 1 \)
• For example, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)
• Note that \( 0! \) is defined as 1, even though 0 is not positive. This is a convention that makes many mathematical formulas work properly.

Knowing how to compute factorials is crucial for solving many types of math problems, including what we see in sequences. It essentially scales up a number by multiplying it by all the smaller numbers down to one.

Substitution Method

The substitution method is a straightforward mathematical technique used to solve sequences and equations. By substituting specific values into a formula, we can calculate individual terms. It helps in breaking down problems step by step. When calculating terms of a sequence, we substitute each consecutive integer starting from 1 for the variable \( n \). Here’s how it works:

• Choose a value for \( n \) — it typically starts at 1 and goes up (1, 2, 3, ...).
• Substitute your chosen \( n \) value into the sequence formula.
• Simplify the expression by following standard operations like addition, subtraction, multiplication, and division.
• Compute the result, which represents a term in the sequence.

Using the substitution method can help you find terms neatly and sequentially without confusion.

Terms of a Sequence

In the context of mathematics, a sequence is a list of numbers arranged in a specific order defined by a formula. Each number in this sequence is called a term. To find these terms, you follow a rule or formula provided for the sequence, like the one given in the exercise:

• The formula for our sequence is \( a_n = \frac{n!}{n^2 - n - 1} \).
• Each term in this sequence is computed by plugging different integer values into \( n \).
• After replacing \( n \) with these integers, we simplify to find specific terms like \( a_1, a_2, a_3, \) etc.

According to the problem, the first four terms are calculated one by one. Understanding the concept of terms and following the formula is essential to grasp sequences in mathematics.