Explainer: Using Arithmetic Sequence Formulas

In this explainer, we will learn how to write explicit and recursive formulas for arithmetic sequences to find the value of the 𝑛th term in an arithmetic sequence and how to find a term’s order given its value.

Definition: Arithmetic Sequences

A sequence is arithmetic if there is a common difference between any two consecutive terms.

An example of an arithmetic sequence would be the two times table: 2,4,6,8,….

Here, there is clearly a common difference of two between each of the consecutive terms. Equally, any other times table is also an arithmetic sequence. Another example would be the sequence of odd numbers: 1,3,5,7,….

This is another sequence with a common difference of two between each consecutive term and can be thought of as the two times table shifted down by one.

Before looking at arithmetic sequences in more detail, let us define some notation. The first term of an arithmetic sequence is generally called π‘Ž or π‘ŽοŠ§. The common difference is generally called 𝑑; the second term, π‘ŽοŠ¨, would then be π‘Ž+𝑑, the third term, π‘ŽοŠ©, would be π‘Ž+2𝑑, and so on:

In order to clarify that a sequence is arithmetic, we would have to check the common difference is equal between every two consecutive terms. To find the common difference, we choose any term from the sequence and then subtract it from the term immediately after it. For example, we could calculate π‘Žβˆ’π‘ŽοŠ¨οŠ§ or π‘Žβˆ’π‘ŽοŠ«οŠͺ. Once we have calculated the common difference, we can easily calculate successive terms of the sequence by adding the difference to the previous term. Let us look at an example. Consider the arithmetic sequence 3,7,11,15,….

We can calculate the common difference 𝑑 by finding the difference between any two terms. Here, consider π‘Ž=3 and π‘Ž=7: 𝑑=π‘Žβˆ’π‘Ž=7βˆ’3=4.

We are given the first four terms, so, to work out the fifth term, π‘ŽοŠ«, we add 4 to the fourth term, π‘ŽοŠͺ: π‘Ž=15+4=19.

The sixth term, π‘ŽοŠ¬, is then π‘Ž+4=23, and finally, π‘Ž=π‘Ž+4=27. So, the next three terms of the sequence are 19,23,27.

For a question like this, you do not need to show this level of working out, but it is useful in developing higher level concepts. For example, we may be asked to find the 𝑛th term of an arithmetic sequence. We have just shown that terms can be calculated by adding the common difference 𝑑 to successive terms. As shown before, the sequence can be represented in the following way:

Notice here that the coefficient of 𝑑 (the number in front of 𝑑) is always one less than the position of the term. The fourth term for instance is π‘Ž+3𝑑. We can generalize this for the 𝑛th term as the coefficient of 𝑑 will be one less than the position, π‘›βˆ’1, so, π‘Ž=π‘Ž+(π‘›βˆ’1)π‘‘οŠ. We can use this formula to calculate the general term or 𝑛th term for any given arithmetic sequence. Let us demonstrate this with an example.

Example 1: Finding the 𝑛th Term of an Arithmetic Sequence

Using the table, determine the expression that represents the value of each term as a function of its position. Then, find the value of the fifteenth term in the sequence.

Position2345𝑛
Value of Term491419

Answer

First, we need to establish the common difference, 𝑑, between each of the consecutive terms. We are told the values of π‘Ž,π‘Ž,π‘ŽοŠ¨οŠ©οŠͺ, and π‘ŽοŠ«, so π‘Žβˆ’π‘Ž=9βˆ’4=5,π‘Žβˆ’π‘Ž=14βˆ’9=5,π‘Žβˆ’π‘Ž=19βˆ’14=5.οŠͺοŠͺ

The common differences are equal so we can assume that the sequence is arithmetic. We know that the second term is equal to π‘Ž+𝑑 where π‘Ž is the first term and 𝑑 is the common difference, so π‘Ž+𝑑=4.

Substituting 𝑑=5, we get π‘Ž+5=4, and solving, we see that π‘Ž=βˆ’1. Recall that the 𝑛th term is π‘Ž=π‘Ž+(π‘›βˆ’1)𝑑.

We can, therefore, substitute the value of π‘Ž and 𝑑 to get π‘Ž=βˆ’1+5(π‘›βˆ’1), which simplifies to π‘Ž=5π‘›βˆ’6.

To find the 15th term, we substitute 𝑛=15: π‘Ž=5(15)βˆ’6=69.

The example above demonstrates a very formal method for finding the 𝑛th term of an arithmetic sequence. We can take a less formal approach which we will demonstrate now.

As we mentioned at the start, a times table is an arithmetic sequence. For example, the two times table: 2,4,6,8,….

If you wanted to find the 7th term of the two times table, you would multiply 2 by 7. If you wanted to calculate the 12th term, you would multiply 2 by 12. Similarly, if you wanted to calculate the 𝑛th term you would multiply 2 by 𝑛 so 2𝑛. For the 8 times table, the 𝑛th term would be 8𝑛 and so on. Any arithmetic sequence can be described as a shift of a times table. Consider the sequence 8,11,14,17,….

Here, the common difference, 𝑑, is 3, so this sequence is a shift of the three times table. We can write down the two sequences and identify the shift as shown below:

We know that the three times table has an 𝑛th term of 3𝑛 and this sequence is a shift of the three times table by +5, so the 𝑛th term of this sequence is 3𝑛+5. We can take a similar approach if the common difference is negative; we just compare the sequence to a negative times table.

Let us look at one more example.

Example 2: Finding the 𝑛th Term of an Arithmetic Sequence

Find, in terms of 𝑛, the general term of the arithmetic sequence βˆ’7,βˆ’5,βˆ’3,βˆ’1,….

Answer

First, we need to calculate the common difference between the terms by subtracting π‘ŽοŠ§ from π‘ŽοŠ¨: π‘Žβˆ’π‘Ž=(βˆ’5)βˆ’(βˆ’7)=2.

We can, therefore, compare the sequence with the two times table as shown below:

We can see from that the sequence is a shift of the two times table down by 9. Hence, the 𝑛th term of the sequence is 2π‘›βˆ’9.

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