Problem 2 - College Algebra with Modeling and Visualization

Question

Write a sequence whose terms represent the first seven positive odd integers.

Step-by-Step Solution

Answer

The sequence is 1, 3, 5, 7, 9, 11, 13.

Step 1: Understand Odd Integers

Odd integers are integers that are not divisible by 2. Typically, they are represented as numbers like 1, 3, 5, etc.

Step 2: Identify the First Positive Odd Integer

The first positive odd integer is 1. This integer is the starting point of our sequence.

Step 3: Add 2 to Generate the Next Odd Integers

Continue the sequence by adding 2 to each subsequent term. Starting from 1, add 2 to get 3, then add 2 to 3 to get 5, and continue this process.

Step 4: Generate the Sequence

Following the addition pattern from Step 3, the sequence is: 1, 3, 5, 7, 9, 11, 13.

Key Concepts

Positive Integers

Positive integers are numbers greater than zero without any decimal or fractional part. These numbers include all the whole numbers starting from 1 upwards (1, 2, 3, and so on). In our current context of odd integers, we only focus on these numbers that meet specific criteria: they are strictly positive and are not divisible by 2.
Some examples of positive integers include:

• 1
• 5
• 20
• 100

This means that when we consider sequences involving positive integers, like odd integers, we only deal with numbers greater than zero.

Sequence Generation

Sequence generation refers to the process of constructing a list of numbers that follow a certain rule. It involves starting with an initial number and then applying a consistent pattern or operation to find the subsequent terms in the sequence. For odd integers, the generation begins with the smallest odd integer, which is 1.
To generate the first few odd integers:

• Begin with 1, the first positive odd integer.
• To get the next term, add 2 to the current term.
• Continue this process to get the sequence: 1, 3, 5, 7, 9, 11, 13.

This method of adding a constant value to reach the next term is a fundamental part of many sequences, especially arithmetic sequences.

Arithmetic Sequences

An arithmetic sequence is a series of numbers with a common difference between consecutive terms. In simpler terms, this means you keep adding (or subtracting) the same number each time to get from one term to the next.
For the sequence of odd integers:

• We start at 1.
• The common difference is 2 (since each term is 2 more than the previous term).
• The sequence follows the pattern as: 1, 3, 5, 7, 9, 11, 13.

Arithmetic sequences can be identified by their uniformity, making them predictable and easily extendable. The formula for finding the nth term of an arithmetic sequence is given by:
\[ a + (n - 1) imes d \]where \( a \) is the first term, \( n \) is the term number, and \( d \) is the common difference. Using this formula, you can easily find any term in the sequence without listing all the terms beforehand.