Problem 43 - College Algebra

Question

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: Identify the formula

We begin by identifying the given formula for the sequence, which is \(a_n = \frac{n!}{n^2}\). This tells us how to calculate each term in the sequence based on the value of \(n\).

Step 2: Calculate the first term

Substitute \(n = 1\) into the formula: \(a_1 = \frac{1!}{1^2} = \frac{1}{1} = 1\). Thus, the first term is 1.

Step 3: Calculate the second term

Substitute \(n = 2\) into the formula: \(a_2 = \frac{2!}{2^2} = \frac{2}{4} = \frac{1}{2}\). Thus, the second term is \(\frac{1}{2}\).

Step 4: Calculate the third term

Substitute \(n = 3\) into the formula: \(a_3 = \frac{3!}{3^2} = \frac{6}{9} = \frac{2}{3}\). Thus, the third term is \(\frac{2}{3}\).

Step 5: Calculate the fourth term

Substitute \(n = 4\) into the formula: \(a_4 = \frac{4!}{4^2} = \frac{24}{16} = \frac{3}{2}\). Thus, the fourth term is \(\frac{3}{2}\).

Key Concepts

Understanding Factorials

A factorial is a mathematical operation that multiplies a number by every positive integer below it. This is denoted with an exclamation mark, as in \(!n\). For example, the factorial of 4, written as \(4!\), is calculated as \(4 \times 3 \times 2 \times 1 = 24\). Factorials grow rapidly as numbers increase and are a key concept in many areas of mathematics.

Factorials are particularly useful in sequences and series, such as in our example sequence where \(n!\) is part of the sequence formula. Understanding how factorials work helps in evaluating sequences accurately. When working with factorials, it's always important to start with the basic value, \(0!\) which is equal to 1, this is a foundation for many calculations involving factorials in more complex sequences.

Sequence Formula Explored

The sequence formula is a mathematical expression that helps us find terms in a sequence. In our given sequence, the formula is \(a_n = \frac{n!}{n^2}\).

This formula tells us how to compute any term of the sequence based on the position \(n\). The numerator of the formula, \(n!\), is the factorial, representing the product of all positive integers up to \(n\). The denominator, \(n^2\), is simply \(n\) squared, or \(n \times n\).

To use the formula:

• Identify the term's position (first term, second term, etc.).
• Substitute the position number for \(n\) in both the numerator and the denominator.
• Simplify the resulting expression to find the term.

This sequence formula combines factorial growth with the simplicity of squaring, providing an interesting balance in calculating terms.

Calculating Terms in the Sequence

To calculate a specific term in our sequence, \(a_n = \frac{n!}{n^2}\), follow a simple substitution and arithmetic process. Each step involves substituting the term number for \(n\) in the formula, calculating the factorial, and then dividing by the square of \(n\). Here is a brief overview:

• First term: Substitute \(n = 1\) into the formula to get \(\frac{1!}{1^2} = 1\).
• Second term: Substitute \(n = 2\) into the formula to get \(\frac{2!}{2^2} = \frac{2}{4} = \frac{1}{2}\).
• Third term: Substitute \(n = 3\) into the formula to get \(\frac{3!}{3^2} = \frac{6}{9} = \frac{2}{3}\).
• Fourth term: Substitute \(n = 4\) into the formula to get \(\frac{4!}{4^2} = \frac{24}{16} = \frac{3}{2}\).

Ensure you simplify fractions where possible to get a clearer understanding of the term's value. This step-by-step calculation helps make each term precise and easy to understand.

Introduction to Mathematical Sequences

Mathematical sequences are ordered lists of numbers which can follow a specific set of rules or patterns defined by a sequence formula. Each number in a sequence is called a 'term'. Understanding sequences is vital as they form the foundation of many mathematical concepts and applications.

In the context of our formula \(a_n = \frac{n!}{n^2}\), this sequence is defined such that each term depends on its position within the sequence. The terms we calculated in the example provide a snapshot of how sequences can behave under given rules.

When analyzing sequences:

• Identify if there is a pattern in how terms are generated.
• Use the pattern or formula to determine missing or subsequent terms.
• Consider the growth or behavior as \(n\) becomes very large, which can lead to insights about the sequence's nature.

Mathematical sequences can vary widely beyond the simple numeric, like arithmetic or geometric sequences, and exploring these can lead to a deeper understanding of mathematical structures.