Explainer: Arithmetic Sequences

In this explainer, we will learn how to calculate the common difference, find next terms in an arithmetic 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: 4,7,10,13,.

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, add 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.

Before we check our understanding with a couple of examples, let’s summarize what we’ve just learned.

Definition: Useful Language

A sequence is an ordered list of numbers.

Each number in the sequence is a term of the 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.

Example 1: Identifying an Arithmetic Sequence

Which of the following is an arithmetic sequence?

  1. (2,10,18,26,)
  2. (6,12,24,48,)
  3. (9,72,144,216,)
  4. 𝑛,𝑛,𝑛,𝑛,
  5. 13,110,117,124,

Answer

In this question, we have five different sequences, and only one of them is an arithmetic sequence. The characteristic of an arithmetic sequence is that the difference between any two consecutive terms is always the same. So, the best way to go ahead is to calculate the difference between the first and second terms, then between the second and third terms, and so on. Then, we will be able to see for which sequence this difference is a constant.

Let’s start with sequence (A). The difference between the second and first terms is 10(2)=10+2=8.

Note that it may be easier to look for the number that needs to be added or subtracted from the first term to get the second term.

Then, to go from 10 to 18, we need again to subtract 8 or add 8.

And 188=26, the value of the fourth term.

We have found that the difference between any two consecutive terms in sequence (A) is 8. Hence, this is an arithmetic sequence whose common difference is 8.

Example 2: Find the Next Terms of an Arithmetic Sequences

Write the next three terms of the arithmetic sequence 161,152,143,134,.

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, 143152. We find that this is 9. Remember that it means that one term is obtained from the previous one by adding 9 (or subtracting 9).

Now that we have found the common difference, we can find the fifth term by subtracting 9 from 134: 1349=125.

The sixth term is then 1259=116.

And the seventh term is 1169=107.

Hence, the next three terms are 125, 116, and 107.

So far, we have learned that an arithmetic sequence is characterized by a constant difference between any two consecutive terms, called the common difference. This characteristic can be used to identify an arithmetic sequence and/or find the next few terms of a given arithmetic sequence. Now, we are going to see how to describe an arithmetic sequence in a more powerful way so that we can find the value of any term of the sequence without having to work out all the previous terms.

Let’s start with the realization that the simplest arithmetic sequences are actually the multiplication tables. Consider the three times table: 1×3=3, 2×3=6, 3×3=9, 4×3=12, and so on. The numbers 3,6,9,12, form an arithmetic sequence with a common difference of 3.

Let’s put these numbers in a table along with the term number.

Term Number1234𝑛
Term Value36912𝑛×3

There is a simple relationship between the term number (this is the position of a term in the sequence, starting here at 1) and the value of the term: 𝑛×3, or 3𝑛. When 𝑛 increases by 1 (when we go from one term to the next one), we see that the value of the term indeed increases by 3, the common difference.

Consider now this arithmetic sequence: 2,1,4,7,. The common difference is 3, as in the previous sequence. The question is, how can we describe the value of any term of the sequence with the term number? Let’s use a diagram with a number line to help us.

The sequence on the top (in blue) is the three times table. The sequence on the bottom (in pink) is the sequence for which we want to find a relationship between a term value and its term number, 𝑛. We observe that because the two sequences have the same common difference (here 3), the bottom one is simply the blue one translated to the left. While the first term in the top one is 3, the first term in the bottom is 2. We can conclude that the second sequence is the first one translated by 5 units to the left. (You can check that it is true for any term number, e.g., the value of the 4th term of the second sequence is indeed the value of the 4th term of the first sequence, 12, minus 5: 125=7.)

The value of the 𝑛th term of the first sequence is described by 3𝑛, so the value of the 𝑛th term in the second sequence is described by 3𝑛5.

This description is now very useful because we can work out the value of any term, simply by plugging in the term number, 𝑛.

Looking back at what we have just done, we realize that the 𝑛th value of any arithmetic sequence will be given by the common difference times 𝑛 plus or minus a number. This is written as 𝑎=𝑑𝑛+𝑐. Here, do not be confused by all these letters! The value of the 𝑛th term is called 𝑎, 𝑑 is the common difference, and 𝑐 is a number (either positive or negative).

Describing the 𝑛th Term of an Arithmetic Sequence

The value of the 𝑛th term of an arithmetic sequence, 𝑎, is mathematically described by 𝑎=𝑑𝑛+𝑐, where 𝑑 is the common difference and 𝑐 is a number (either positive or negative).

Let’s look at some further examples together.

Example 3: Finding the First Terms of an Arithmetic Sequence Using the Description of Its 𝑛th Term

Find the first five terms of the sequence whose general term is given by 𝑎=4𝑛+1 where 𝑛1.

Answer

We are given the mathematical description of the value of the 𝑛th term 𝑎: 𝑎=4𝑛+1, where 𝑛 is the term number, starting at 1 since it is said that 𝑛1. We are going to use this description to find the first five terms of the sequence.

The first term is when 𝑛=1, so, we need to replace 𝑛 in the equation with 1 to find 𝑎: 𝑎=4×1+1=5.

Similarly, we find that the second term 𝑎 is given by 𝑎=4×2+1=9 and the following terms: 𝑎=4×3+1=13,𝑎=4×4+1=17,𝑎=4×5+1=21.

Note that we could have also recognized straight away that it is an arithmetic sequence since the value of its 𝑛th term in terms of 𝑛 is in the form 𝑎=𝑑𝑛+𝑐. Comparing with 𝑎=4𝑛+1, we conclude that the common difference is 4. From this, it follows that we can derive the four next terms, once we have found the value of 𝑎, by adding 4 to one term to get the next one.

Hence, the first five terms are 5, 9, 13, 17, 21.

Example 4: Describing the 𝑛th Term of an Arithmetic Sequence and Using It to Find a Term

Jacob started his action figure collection, where each year he buys 8 action figures.

Write an algebraic expression that can be used to find the number of action figures he would have after 𝑛 years, and then determine how many action figures he would have after 24 years.

Answer

Here, we want to mathematically describe the number of action figures Jacob would have after 𝑛 years as described with words in the question.

Let’s analyze the description given in the question. Jacob is collecting action figures, and we read that he buys 8 of them each year. This means that, in the first year, he will have 8 figures, in the second year 16, in the third year 24, and so on. Let’s put these numbers in a table to help us.

Year123
Number of Action Figures81624

In other words, each year the number of Jacob’s action figures increases by 8; this corresponds to an arithmetic sequence of common difference 8. The value of the 𝑛th term of an arithmetic sequence, 𝑎, is mathematically described by 𝑎=𝑑𝑛+𝑐, where 𝑑 is the common difference and 𝑐 is a number (either positive or negative).

We find here that the number of action figures in the 𝑛th year is 𝑎=8𝑛.

To find how many action figures Jacob would have after 24 years, we simply need to replace 𝑛 in our equation by 24: 𝑎=8×24=192.

So, Jacob would have 192 action figures in 24 years.

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