**Identify arithmetic and geometric sequences** *KS3 Revision*

## What you need to know

**Things to remember:**

- If we add or subtract the same amount each time, the sequence is arithmetic.
- If we multiply or divided by the same amount each time, the sequence is geometric.

Looking at the following two sequences what differences can we spot by?

-2 -4 -6 -8 -10 …

3 9 27 81 243…

- The first sequence is even, and the second sequence is odd.
- The first sequence is negative, and the second sequence is positive.
- The first sequence is going down, the second sequence is going up.
- The first sequence we subtract 2 each time, the second sequence we multiply by 3 each time.

Point 4 is the important one, whether we are adding/subtracting or if we are multiplying/dividing. If we are adding or subtracting our sequence is arithmetic, but if we are multiplying or dividing, the sequence is geometric. So, to determine whether a sequence is arithmetic or geometric, we need to do two things:

**Arithmetic check: **Subtract pairs of terms that are next to each other. If the difference is the same, it is arithmetic.

**Geometric check: **Divide pairs of terms that are next to each other. If the answer is the same, it is geometric.

*Is the following sequence arithmetic or geometric?*

* *

13 17 21 25 29 …

Arithmetic check

** **

17-13=4

21-17=4

25-21=4

Our differences are the same, so the sequence is arithmetic!

We don’t need to check if it is geometric, but let’s do it anyway to see what happens.

**Geometric check**

** **

17\div13=1.\overline{307962}

21\div17=1.23529…

25\div21=1.\overline{190476}

** **

We can see that all of these differences are different, so it isn’t geometric.

*Is the following sequence arithmetic or geometric?*

5 15 45 135 405 …

**Arithmetic check**

** **

15-5=10

45-15=30

135-45=90

We can see that these differences are all different, so the sequence can’t be arithmetic.

**Geometric check**

** **

15\div5=3

45\div15=3

135\div45=3

All of the differences are 3, so this sequence is arithmetic.

## Example Questions

**Question 1:** Is the following sequence arithmetic or geometric?

3 12 48 192 768 …

**Arithmetic check**

** **

12-3=9

48-12=36

192-48=144

We can see that these differences are all different, so the sequence can’t be arithmetic.

**Geometric check**

** **

12\div3=4

48\div12=4

192\div48=4

All of the differences are 4, so this sequence is arithmetic.

**Question 2:** Is the following sequence arithmetic or geometric?

2 11 20 29 38 …

**Arithmetic check **

** **

11-2=9

20-11=9

29-20=9

Our differences are the same, so the sequence is arithmetic!