Identify arithmetic and geometric sequences

Key notes:

In an arithmetic sequence, there is a constant difference between consecutive terms. This means that you can always get from one term to the next by adding or subtracting the same number.

In a geometric sequence, there is a constant multiplier between consecutive terms. This means that you can always get from one term to the next by multiplying or dividing by the same number.

The only way a sequence can be both arithmetic and geometric is if it repeats the same number over and over again.

Learn with an example

👉 What kind of sequence is this?

16, 25, 36, 49, …

  • arithmetic
  • geometric
  • both
  • neither

First check if the sequence is arithmetic. There is not a constant difference between consecutive terms. So, the sequence is not arithmetic.

Next check if the sequence is geometric. There is not a constant multiplier between consecutive terms. So, the sequence is not geometric.

These multipliers are not all the same:

25 / 16  ≈ 1.563

36 / 25 = 1.44

49 / 36 ≈ 1.361

So, the sequence is neither arithmetic nor geometric.

👉 On his mobile phone plan, Henry used 186 minutes in October, 198 minutes in November, 209 minutes in December, and 219 minutes in January. What kind of sequence is this?

  • arithmetic
  • geometric
  • both
  • neither

First check if the sequence is arithmetic. There is not a constant difference between consecutive terms. So, the sequence is not arithmetic.

Next check if the sequence is geometric. There is not a constant multiplier between consecutive terms. So, the sequence is not geometric.

These multipliers are not all the same:

198 / 186 ≈ 1.065

209 / 198 ≈ 1.056

219 / 209 ≈ 1.048

So, the sequence is neither arithmetic nor geometric.

👉 What kind of sequence is this?

78, 87, 96, 105, …

  • arithmetic
  • geometric
  • both
  • neither

First check if the sequence is arithmetic. There is a constant difference of 9 between consecutive terms. So, the sequence is arithmetic.

Next check if the sequence is geometric. There is not a constant multiplier between consecutive terms. So, the sequence is not geometric.

These multipliers are not all the same:

87 / 78 ≈ 1.115

96 / 87 ≈ 1.103

105 / 96 ≈ 1.094

So, the sequence is only arithmetic.

Let’s practice!🖊️