Question

How do you determine whether each sequence is an arithmetic sequence: 18, 16, 15, 13,...?

Answer 1

This sequence is not arithmetic; the difference between successive terms changes.

Explanation:

The differences (or "gaps", if you will) between numbers in an arithmetic sequence are constant. This sequence is not an arithmetic sequence, because `18` and `16` differ by `2`, while `16` and `15` differ by only `1`. The terms do not have equal spacing, and so the series is not arithmetic.

The general way to test this for any sequence `a_1,a_2,a_3,...` is to check if `(a_2-a_1)=(a_3-a_2)=(a_4-a_3)=...=(a_n-a_(n-1))`. If any of these equalities is invalid, the sequence is not arithmetic.