Problem 27 - Algebra for College Students

Question

Write a formula for the general term of each infinite sequence. \(1,3,5,7,9, \dots\)

Step-by-Step Solution

Answer

The general term is a_n = 2n - 1.

Step 1: Identify the Pattern

First, identify the pattern in the sequence. The given sequence is: 1, 3, 5, 7, 9, ... Look at the differences between each consecutive term: 3 - 1 = 2, 5 - 3 = 2, 7 - 5 = 2, 9 - 7 = 2. The common difference is 2.

Step 2: Determine the First Term

The first term of the sequence is 1. This is denoted as: a_1 = 1.

Step 3: Use the Arithmetic Sequence Formula

The formula for the general term of an arithmetic sequence is given by: a_n = a_1 + (n - 1)d, where a_n is the general term, a_1 is the first term, d is the common difference, n is the term number.

Step 4: Substitute the Known Values

Now, substitute the known values into the arithmetic sequence formula. Here, a_1 = 1 and d = 2. a_n = 1 + (n - 1)2.

Step 5: Simplify the Expression

Simplify the expression to find the general term: a_n = 1 + 2n - 2 = 2n - 1.

Key Concepts

arithmetic sequence

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is known as the 'common difference.'
The sequence in the exercise, 1, 3, 5, 7, 9, ..., is an example of an arithmetic sequence, because the difference between each term is always 2.
Arithmetic sequences are very useful in various mathematical applications and are fundamental to understanding more complex topics in arithmetic and algebra. Identifying an arithmetic sequence involves checking if the differences between consecutive terms are equal. If they are, then you have an arithmetic sequence!

common difference

The common difference, denoted as \(d\), is the value that separates consecutive terms in an arithmetic sequence. For the sequence 1, 3, 5, 7, 9, ..., the common difference is 2, since each term increases by 2.
To find the common difference, subtract any term from the next term. For example:

• 3 - 1 = 2
• 5 - 3 = 2
• 7 - 5 = 2

These results show that the common difference is consistent and equals 2. Understanding the common difference is crucial for constructing the general term formula of an arithmetic sequence.

sequence formula

The general term of an arithmetic sequence can be found using the formula: \[ a_n = a_1 + (n - 1) \times d \]where:

• \(a_n\) is the general term.
• \(a_1\) is the first term.
• \(d\) is the common difference.
• \(n\) is the term number.

To apply this to the sequence 1, 3, 5, 7, 9, ..., we first identify the first term, which is 1 (\(a_1 = 1\)), and the common difference, which is 2 (\(d = 2\)).Substituting these values into the formula gives us:\[ a_n = 1 + (n - 1) \times 2 \]Simplifying this:\[ a_n = 1 + 2n - 2 \]\[ a_n = 2n - 1 \]So, the general term formula for this sequence is \(2n - 1\). This formula allows you to find any term in the sequence by plugging in the desired term number (\(n\)). For example, to find the 5th term (\(a_5\)):\[ a_5 = 2 \times 5 - 1 = 10 - 1 = 9 \]