Problem 10 - Beginning and Intermediate Algebra

Question

If the given sequence is arithmetic, find the common difference \(d .\) If the sequence is not arithmetic, say so. See Example 1. \(2,5,8,11, \ldots\)

Step-by-Step Solution

Answer

The sequence is arithmetic with a common difference of 3.

Step 1 - Identify the Sequence Type

First, we need to determine whether the given sequence is arithmetic. An arithmetic sequence has a constant difference between consecutive terms.

Step 2 - Calculate the Differences

Calculate the differences between consecutive terms. For the given sequence: 2, 5, 8, 11,... The differences are: 5 - 2 = 3 8 - 5 = 3 11 - 8 = 3

Step 3 - Verify the Common Difference

Verify that the differences calculated are the same. Since all differences are equal to 3, the sequence is arithmetic and the common difference is 3.

Key Concepts

common difference

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This difference is called the common difference, denoted as d.
To find the common difference, you subtract any term in the sequence from the term that follows it. In the given exercise, we calculate it as follows:

• 5 - 2 = 3
• 8 - 5 = 3
• 11 - 8 = 3

Since the difference between each pair of consecutive terms (2 and 5, 5 and 8, 8 and 11) is 3, the common difference d is 3.

sequence types

A sequence is a set of numbers arranged in a specific order. There are different types of sequences, but the most common types are arithmetic and geometric sequences.

• Arithmetic Sequence: This sequence has a constant difference, called the common difference, between each pair of consecutive terms. For example, in the sequence 2, 5, 8, 11,..., the common difference is 3.
• Geometric Sequence: This sequence has a constant ratio, called the common ratio, between each pair of consecutive terms. For example, in the sequence 2, 6, 18, 54,..., the common ratio is 3 (each term is multiplied by 3 to get the next term).

Identifying the type of sequence is crucial because it determines how you will solve problems related to it. In this exercise, recognizing the sequence as arithmetic was the first step.

consecutive terms

Consecutive terms in a sequence are terms that come one after another, without any gaps between them. For example, in the sequence 2, 5, 8, 11,... the terms 2 and 5 are consecutive, as are 5 and 8, and 8 and 11.
When analyzing sequences, especially arithmetic ones, checking the differences between consecutive terms helps to determine if the sequence has a constant difference. In this exercise:

• The difference between the consecutive terms 5 and 2 is 3.
• The difference between 8 and 5 is 3.
• The difference between 11 and 8 is 3.

Since these differences are constant, it confirms that we have an arithmetic sequence.