In this explainer, we will learn how to calculate the common difference in an arithmetic sequence, find subsequent terms in the sequence, and check if the sequence increases or decreases.
Let’s look at an example. Consider this pattern and count the number of lines for each diagram.
We find that there are 4 lines for the first diagram, 7 for the second, 10 for the third, and 13 for the fourth. Let’s use these numbers of lines to form a sequence:
We add an ellipsis here (the three dots at the end) to indicate that the sequence can go on forever.
Let’s analyze this sequence, and in particular, let’s find out how each term in the sequence is obtained from the previous one. We see that the second term, 7, is the first term, 4, added by 3; then, we need again to add 3 to 7 (second term) to get 10 (third term) and the same to go from 10 (third term) to 13 (fourth term):
This characteristic of our sequence can be described by saying that the difference between any two consecutive terms is always the same. This defines an arithmetic sequence. And because the difference between two consecutive terms is the same whatever the two consecutive terms we pick, it is called the common difference.
Let’s formally define these terms.
Definition: Arithmetic Sequence
An arithmetic sequence is a sequence in which the difference between any two consecutive terms is always the same. The difference between any two consecutive terms of an arithmetic sequence is called the common difference.
Let’s begin with an example where we will determine which of the given sequences is arithmetic.
Example 1: Identifying Arithmetic Sequences
Which of the following is an arithmetic sequence?
Answer
In this example, we need to determine which of the given sequences is arithmetic. Recall that an arithmetic sequence is a sequence in which the difference between any two consecutive terms is always the same. In order to determine whether a given sequence is arithmetic, we need to make sure that the difference between each consecutive term is the same. When taking the difference, we subtract the previous term from the next term.
Let’s check this condition for each sequence. We can stop computing the differences when we notice that the difference we computed is not the same as the one we computed previously.
For the sequence in A, the difference between the second and first term is however, the difference between the third and second term is
While we can compute another difference, , it is not necessary to do so since we can already see that the two differences we computed, and 36, are not equal. Hence, this sequence is not arithmetic.
Next, consider the differences between the first few terms of the sequence in B:
Since , this sequence is not arithmetic.
The sequence in C is given in terms of an unknown constant . We can see that this sequence is generated by multiplying, rather than adding, the previous term by . In this case, consecutive terms have a common multiple, not a common difference. Hence, this sequence is not arithmetic.
For the sequence in D, it is more difficult to compute the differences since each term is a fraction. We can see that each term is generated by adding 7 to the denominator of the fraction. However, adding a constant to the denominator is different from adding a constant to the number. In particular, the difference between each consecutive pair of terms would not be equal. Hence, this sequence is not arithmetic.
Finally, let’s consider the sequence in E. The differences between each of the consecutive terms are
We can see that the difference between each consecutive term is equal to . Hence, this sequence is arithmetic.
The only arithmetic sequence is option E.
In the previous example, we identified an arithmetic sequence from a collection of sequences by identifying the difference between each two consecutive terms. Recall that this difference, which must be a constant for an arithmetic sequence, is called the common difference. If we know the common difference of an arithmetic sequence, we can find the next term in the arithmetic sequence by adding the common difference to the previous term. For instance, recall the arithmetic sequence discussed earlier:
We noted that the common difference of this arithmetic sequence is equal to 3. We can find the term that comes directly after 13 in this sequence by
We can continue this process to generate as many terms as we need.
In the next example, we will find the next three terms of an arithmetic sequence.
Example 2: Finding Consecutive Terms of an Arithmetic Sequence
Write the next 3 terms in the arithmetic sequence 12, 19, 26, 33, .
Answer
In this example, we need to find the next 3 terms of a given arithmetic sequence. Recall that an arithmetic sequence is a sequence in which the difference between any two consecutive terms is always the same. The difference between any two consecutive terms of an arithmetic sequence is called the common difference. To find the next term in the arithmetic sequence, we can add the common difference to the previous term. Let’s begin by finding the common difference.
To find the common difference, we can find the difference of any two consecutive terms. For instance, the difference between the first two terms is given by
This tells us that the common difference of this arithmetic sequence is equal to 7. We can verify this by checking whether the next term can be obtained by adding 7 to the previous term:
We can see that adding 7 to the second term produces the third term, and adding 7 to the third term produces the fourth term. This verifies that 7 is the common difference.
We can find the fourth, fifth, and sixth term of this sequence by consecutively adding 7 to the previous term. We will use the notation to represent the term of this sequence. We can compute
This tells us that the next 3 terms in the arithmetic sequence are
The common difference of the arithmetic sequence in the previous example is positive, and we can see that each term of the sequence is larger than the previous term. This tells us that this sequence is increasing.
Definition: Common Difference of an Arithmetic Sequence
For ,, we denote to be the term of an arithmetic sequence. For any , the common difference of the arithmetic sequence is given by
Equivalently, if we are given the common difference, we can write the next term of the arithmetic sequence as
We can use the definition above to understand why an arithmetic sequence is increasing if the common difference is positive. If the common difference, , of an arithmetic sequence is positive, we can write
Since the left-hand side of this inequality is equal to the next term, , in the arithmetic sequence, this implies , which means that the arithmetic sequence is increasing.
Similarly, if the common difference is negative, we can use a similar argument to write
This tells us
Hence, the arithmetic sequence is decreasing if the common difference is negative.
Definition: Increasing and Decreasing Arithmetic Sequences
An arithmetic sequence is increasing if for all . This happens when the common difference is positive.
An arithmetic sequence is decreasing if for all . This happens when the common difference is negative.
In the next example, we will determine whether the given arithmetic sequence is decreasing.
Example 3: Discussing the Monotonicity of an Arithmetic Sequence
Is the sequence increasing or decreasing?
Answer
In this example, we need to determine whether the given sequence is increasing or decreasing. We will demonstrate two different methods to answer this question.
Method 1
Recall that an arithmetic sequence is increasing or decreasing if its common difference is positive or negative respectively. We can find the common difference by subtracting a term in the arithmetic sequence from the next term. For instance, taking the difference of the first two terms gives us
Hence, the common difference of this arithmetic sequence is . Since the common difference is negative, this sequence is decreasing.
Method 2
We can also determine whether a sequence is increasing or decreasing by comparing the size of consecutive terms. We can see that each term in the sequence is smaller than the previous term:
Hence, the sequence is decreasing.
We can also find missing terms in a decreasing arithmetic sequence by adding the common difference, which is negative, to the previous term. Let’s compute the next 3 terms in a decreasing arithmetic sequence.
Example 4: Finding the Next Terms of an Arithmetic Sequences
Write the next three terms of the arithmetic sequence .
Answer
In this question, we are looking at an arithmetic sequence. The characteristic of an arithmetic sequence is that the difference between any two consecutive terms is always the same. We are going to use this fact to find the next three terms.
But first, we need of course to find this difference, called the common difference. For this, we simply need to work out the difference between one term and the previous term, for instance, . We find that this is . Remember that this means that one term is obtained from the previous one by adding (or subtracting 9).
Now that we have found the common difference, we can find the fifth term by subtracting 9 from 134:
The sixth term is then
And the seventh term is
Hence, the next three terms are 125, 116, and 107.
In the final example, we will find an unknown constant by using the property of arithmetic sequences.
Example 5: Using a Given Arithmetic Sequence to Form and Solve a Linear Equation
Find given three consecutive terms of an arithmetic sequence are , and .
Answer
In this example, we need to find the unknown constant where expressions involving this constant form consecutive terms of an arithmetic sequence. Recall that an arithmetic sequence is a sequence in which the difference between any two consecutive terms is always the same.
Let’s compute the differences of consecutive terms. There are two such differences to compute:
Since these differences must be equal, we have
Rearranging this equation,
Let’s finish by recapping a few important concepts from this explainer.
Key Points
- An arithmetic sequence is a sequence in which the difference between any two consecutive terms is always the same. The difference between any two consecutive terms of an arithmetic sequence is called the common difference.
- If the term of an arithmetic sequence is denoted , the common difference, , of this arithmetic sequence is given by
- We can find the next term in the arithmetic sequence by adding the common difference to the previous term.
- If the common difference of an arithmetic sequence is positive, the sequence is increasing. Likewise, if the common difference of an arithmetic sequence is negative, the sequence is decreasing.