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!