what is arithmetic sequence

2 min read 08-03-2025

An arithmetic sequence, also known as an arithmetic progression, is a mathematical sequence where the difference between consecutive terms is constant. This constant difference is called the common difference. Understanding arithmetic sequences is fundamental to algebra and has applications in various fields. This article will delve into the definition, properties, and examples of arithmetic sequences.

Understanding the Core Concept: The Common Difference

The defining characteristic of an arithmetic sequence is its common difference. Let's say we have a sequence: 2, 5, 8, 11, 14...

To find the common difference, subtract any term from the term that immediately follows it. For example:

5 - 2 = 3
8 - 5 = 3
11 - 8 = 3
14 - 11 = 3

The common difference in this sequence is 3. This means each term is obtained by adding 3 to the previous term. This consistency is what makes it an arithmetic sequence.

Key Properties of Arithmetic Sequences

Several properties help us identify and work with arithmetic sequences:

Constant Difference: As mentioned, the most important characteristic is the consistent difference between consecutive terms.
Linear Relationship: Arithmetic sequences represent a linear relationship. If you were to graph the terms against their position in the sequence, you'd get a straight line.
Formula for the nth Term: We can find any term in the sequence without listing them all out. The formula is: an = a1 + (n-1)d
- Where:
 - an is the nth term in the sequence
 - a1 is the first term
 - n is the term number
 - d is the common difference

Examples of Arithmetic Sequences

Let's look at a few more examples to solidify our understanding:

Example 1: 1, 4, 7, 10, 13... (Common difference: 3)
Example 2: 10, 7, 4, 1, -2... (Common difference: -3 – note that the common difference can be negative)
Example 3: -5, -2, 1, 4, 7... (Common difference: 3)

Notice that the terms don't have to be positive integers; they can be negative or include decimals/fractions as well, as long as the common difference remains constant.

How to Identify an Arithmetic Sequence

To determine if a sequence is arithmetic, simply calculate the difference between consecutive terms. If the difference is constant, it's an arithmetic sequence. If the difference varies, it's not an arithmetic sequence.

Finding the nth Term: A Worked Example

Let's use the formula to find the 10th term (a₁₀) of the sequence: 2, 5, 8, 11…

Identify the first term (a₁): a₁ = 2
Identify the common difference (d): d = 3
Use the formula: a_n = a₁ + (n-1)d
Substitute and solve: a₁₀ = 2 + (10-1)3 = 2 + 27 = 29

Therefore, the 10th term of this sequence is 29.

Applications of Arithmetic Sequences

Arithmetic sequences appear in various real-world scenarios:

Simple Interest: The yearly balance in a savings account with simple interest forms an arithmetic sequence.
Linear Growth: Many scenarios involving linear growth, like the increase in population at a constant rate, can be modeled using an arithmetic sequence.
Physics: Problems involving uniformly accelerated motion often involve arithmetic sequences.

Conclusion

Arithmetic sequences are a fundamental concept in mathematics. Understanding their properties, especially the common difference and the formula for the nth term, allows you to solve various problems and model real-world situations effectively. By mastering this concept, you’ll build a stronger foundation in algebra and related fields. Remember, the key is that consistent common difference between consecutive terms!