When a sequence is arithmetic, the difference between two consecutive terms is constant. This difference is called a common difference because it is common between each pair of terms. Examining our sequence, we can see that this is not the case.

18 \to - 2 16 \to - 1 15 \to - 2 13

Because the difference between the terms is not the same, we know that the sequence is not arithmetic.

Extra

Other Types of Sequences

Not all sequences are arithmetic. The most frequently seen type of non-arithmetic sequence is called a geometric sequence. In a geometric sequence, there is a common ratio instead of a common difference. The common ratio either multiplies the terms by the same number each time. The multiplicative factor can be an integer.

2 \to \times 2 4 \to \times 2 8 \to \times 2 16

Or the factor can be a fraction.

27 \to \times \frac{1}{3} 9 \to \times \frac{1}{3} 3 \to \times \frac{1}{3} 1

There are also sequences that follow a rule, rather than have a common difference or ratio. For example, in the following sequence the rule is

n^{2} + 1

where

n

is the number of the term in the sequence.

Term : Rule : Sequence : n = 1 1^{2} + 1 2 \to n = 2 2^{2} + 1 5 \to n = 3 3^{2} + 1 10

If we were to continue the given sequence, we might see that the pattern is to alternate between subtracting

2

and subtracting

1 .

This would be a sequence that followed a rule, just not an arithmetic one.

18 \to - 2 16 \to - 1 15 \to - 2 13 \to - 1 12 \to - 2 10

Exercise