How do you tell whether the sequence -1,1,3,5,.... is arithmetic, geometric or neither?

1 Answer
Jun 8, 2018

Answer:

The sequence is arithmetic

Explanation:

A sequence #a_n# is:

  • Arithmetic if the difference between two consecute elements, i.e. #a_n-a_{n-1}#, is constant.
  • Geometric if the ratio between two consecute elements, i.e. #a_n/a_{n-1}#, is constant.

Let's test both to get the answer.

Difference test
Let's consider the difference between every element and the following one:

  • #a_2-a_1= 1-(-1)=1+1=2#
  • #a_3-a_2 = 3-1=2#
  • #a_4-a_3 = 5-3=2#

The difference remains constant (it's always #2#), so we may already conclude that the sequence is arithmetic. Let's do the ratio test nevertheless:

Ratio test

Let's consider the ratio between every element and the following one:

  • #a_2/a_1= 1/(-1)=-1#
  • #a_3/a_2 = 3/1=3#
  • #a_4/a_3 = 5/3=1.666...#

The ratio is always different, so the sequence is not geometric, as expected.