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Arithmetic Sequences

KS3 Maths Algebra Arithmetic Sequences

In maths, sequences are like lists of numbers that follow a set rule. Out of these sequences, arithmetic sequences are perhaps the most straightforward to understand.

In an arithmetic sequence, you simply add the same number to each term to get the next one. This fixed number you add is called the common difference.

Identifying Arithmetic Sequences

Recognising an arithmetic sequence is easy. If you can get from one term to the next by adding the same number every time, you’ve identified an arithmetic sequence.

For example, in the sequence 2, 4, 6, 8, you’re adding 2 each time, so it’s an arithmetic sequence.

In contrast, the sequence 1, 2, 4, 8 is not an arithmetic sequence because the difference between the terms is not constant.

Common Difference

The common difference is the heartbeat of an arithmetic sequence. It’s the number that you add to each term to get the next one. In the sequence 3, 5, 7, 9, the common difference is 5 – 3 = 2.

Let’s look at some more examples.

Identifying the common difference

In the sequence:

  • 10, 15, 20, 25, the common difference is15 – 10 = 5
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    -2, 0, 2, 4, the common difference is0 – (-2) = 2
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. Even though the sequence includes negative numbers, the concept of a common difference still applies.</mark></li>
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    100, 90, 80, 70, the common difference is90 – 100 = -10
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. Notice that common differences can also be negative, which means each subsequent term is smaller than the previous one.</mark></li>
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    \frac{1}{2}, \frac{3}{2}, \frac{5}{2}, \frac{7}{2}, the common difference is(\frac{3}{2}) – (\frac{1}{2}) = 1
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. This shows that common differences can also be fractions.</mark></li>
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    1, 4, 7, 10, 13, the common difference is4 – 1 = 3, meaning each term increases by 3 to get to the next term

Finding the nth Term in an Arithmetic Sequence

If you know the first term and the common difference, you can find any term in the sequence. For example, if the first term is 4 and the common difference is 3, the third term will be 4 + 3 + 3 = 10.

Let’s look at some more examples:

  • If the first term is 5 and the common difference is -2, the fourth term will be calculated as 5 + (-2) + (-2) + (-2) = -1. Or 
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5 + (3 \times -2)</mark></li>
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<li><mark style="background-color:rgba(0, 0, 0, 0)" class="has-inline-color has-black-color">If the first term is \frac{1}{2} and the common difference is \frac{1}{2}, the fifth term will be

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    \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} = \frac{5}{2}. Or,\frac{1}{2} \times 5.
  • If the first term is 0 and the common difference is 4, the second term will simply be 0 + 4 = 4.

Practice Questions

Question 1:

Find the common difference of the sequence 3, 6, 9

The common difference is 6 - 3 = 3

Question 2:

What is the 5th term in the sequence 2, 5, 8?

The common difference is 5 - 2 = 3

Add the common difference to the 3rd term (8) twice to get the 5th term: 8 + 3 + 3 = 14

Question 3:

Is the sequence 1, 4, 9 an arithmetic sequence?

No, it is not an arithmetic sequence

The differences between terms are not constant (4 - 1 = 3, 9 - 4 = 5)

Question 4:

Find the 7th term in the sequence 1, 3, 5, 7

The common difference is 3 - 1 = 2

Add the common difference to the 4th term (7) three times to get the 7th term: 7 + 2 + 2 + 2 = 13

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