Problem 43 - College Algebra with Corequisite Support

Question

For the following exercises, write the first four terms of the sequence. $$ a_{n}=\frac{n !}{n^{2}} $$

Step-by-Step Solution

Answer

The first four terms are 1, \( \frac{1}{2} \), \( \frac{2}{3} \), \( \frac{3}{2} \).

Step 1: Define the formula for the sequence

The given formula for the sequence is \( a_{n} = \frac{n!}{n^2} \). We will use this formula to calculate the first four terms of the sequence by substituting different values of \( n \).

Step 2: Calculate the first term \( a_1 \)

Substitute \( n = 1 \) into the formula: \[ a_1 = \frac{1!}{1^2} = \frac{1}{1} = 1 \]

Step 3: Calculate the second term \( a_2 \)

Substitute \( n = 2 \) into the formula: \[ a_2 = \frac{2!}{2^2} = \frac{2}{4} = \frac{1}{2} \]

Step 4: Calculate the third term \( a_3 \)

Substitute \( n = 3 \) into the formula: \[ a_3 = \frac{3!}{3^2} = \frac{6}{9} = \frac{2}{3} \]

Step 5: Calculate the fourth term \( a_4 \)

Substitute \( n = 4 \) into the formula: \[ a_4 = \frac{4!}{4^2} = \frac{24}{16} = \frac{3}{2} \]

Step 6: List the first four terms

The first four terms of the sequence based on our calculations are: \( 1, \frac{1}{2}, \frac{2}{3}, \frac{3}{2} \).

Key Concepts

Factorials

Factorials are mathematical expressions that are very useful in sequences and various algebraic calculations. A factorial, denoted by an exclamation mark (!), is the product of all positive integers less than or equal to a number. For example:

• The factorial of 1, which is written as \(1!\), is simply 1, because there are no positive integers less than 1 to multiply with.
• \(2! = 2 \times 1 = 2\).
• \(3! = 3 \times 2 \times 1 = 6\).
• \(4! = 4 \times 3 \times 2 \times 1 = 24\).

As we can see, a factorial grows rapidly with larger numbers. This characteristic is useful in permutations and combinations where we need to calculate possible arrangements or selections.

Mathematical Sequences

Mathematical sequences are ordered lists of numbers that follow a particular rule or pattern. In this context, each term in the sequence can be calculated using a formula.For example, the formula given in the exercise is\[a_{n} = \frac{n!}{n^{2}}\]This specific sequence is determined by plugging in different values of \(n\) (like 1, 2, 3, or 4) into the formula, which helps us find the terms in the sequence.
There's a clear procedure here:

• Identify the formula.
• Substitute each consecutive integer for \(n\).
• Simplify the expression to find each respective term.

Sequences can be arithmetic, geometric, or like this one, defined by a unique formula involving factorials and powers.

Algebraic Expressions

Algebraic expressions are mathematical statements that combine numbers, variables, and operations. They can contain addition, subtraction, multiplication, division, and even factorials.In our exercise, the expression \(\frac{n!}{n^2}\) involves both factorials and polynomial terms, specifically involving squares. This combination makes the sequence distinct and gives it interesting properties.
Key aspects to keep in mind when working with algebraic expressions include:

• Understanding each part of the expression such as \(n^2\) which represents \(n\) squared.
• Simplifying expressions through factorization or cancellation, especially when dealing with fractions.
• Identifying patterns or behaviors as \(n\) changes, which is crucial in finding terms of the sequence.

Learning to manipulate algebraic expressions is foundational in algebra and important when working with sequences.