# Arithmetic and Geometric Sequences The two main types of series/sequences are arithmetic and geometric. Some sequences are neither of these. It's important to be able to identify what type of sequence is being dealt with. An arithmetic series is one where each term is equal the one before it plus some number. For example: 5, 10, 15, 20,… Each term in this sequence equals the term before it with 5 added on.

In contrast, a geometric sequence is one where each term equals the one before it multiplied by a certain value. An example would be 3, 6, 12, 24, 48,… Each term is equal to the prior one multiplied by 2. Some sequences are neither arithmetic nor geometric. An example would be 1, 2, 3, 2, 1, 2, 3, 2, 1,… The terms in this sequence all differ by 1, but sometimes 1 is being added and other times it is being subtracted, so the sequence is not arithmetic. Also, there is no common value being multiplied by one term to get the next, so the sequence cannot be geometric, either. Arithmetic sequences grow very slowly in comparison with geometric sequences.

### Try Identifying What Type of Sequences Are Shown Below

1. 2, 4, 8, 16,…

2. 3, -3, 3, -3,…

3. 1, 2, 3, 4, 5, 6, 7,…

4. -4, 1, 6, 11, 16,…

5. 1, 3, 4, 7, 8, 11,…

6. 9, 18, 36, 72,…

7. 7, 5, 6, 4, 5, 3,…

8. 10, 12, 16, 24,…

9. 9, 6, 3, 0, -3, -6,…

10. 5, 5, 5, 5, 5, 5,…

### Solutions

1. Geometric with common ratio of 2

2. Geometric with common ratio of -1

3. Arithmetic with common value of 1

4. Arithmetic with common value of 5

5. Neither geometric nor arithmetic

6. Geometric with common ratio of 2

7. Neither geometric nor arithmetic

8. Neither geometric nor arithmetic

9. Arithmetic with common value of -3

10. Either arithmetic with common value of 0 or geometric with common ratio of 1