Whether the sequence -1,1,3,5,.... is arithmetic, geometric or neither

Colten Andrade

Colten Andrade

Answered question

2022-09-26

Whether the sequence -1,1,3,5,.... is arithmetic, geometric or neither

Answer & Explanation

Lilia Horton

Lilia Horton

Beginner2022-09-27Added 6 answers

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.

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