# How do you determine if 15,-5,-25,-45 is an arithmetic or geometric sequence?

Jun 27, 2015

The sequence is an arithmetic sequence.

#### Explanation:

• In an Arithmetic sequence there is a common difference $d$ between any two consecutive terms
• In a Geometric sequence there is a common ratio $r$ for any two consecutive terms

The sequence given is :

$15 , - 5 , - 25 , - 45$

1) Checking if the sequence is an arithmetic sequence:

 color(blue)(d _1= a_2 - a_1) = -5 - (15) = color(blue)(-20
 d_2 = a_3 - a_2 = -25 - (-5) =color(blue)( -20
 d_3 = a_4 - a_3= -45 - (-25) = color(blue)(-20

As observed ${d}_{1} = {d}_{2} = {d}_{3}$ , so there is a common difference $d = - 20$ maintained in the sequence so it is an arithmetic sequence

2) Checking if the sequence is also a geometric sequence:

 color(blue)(r _1= a_2/ a_1) = -5 / 15 = color(blue)(-1/3

 color(blue)(r _2 = a_3/ a_2) = (-25 )/ -5 = color(blue)(5

Since ${r}_{1}$ is not equal to ${r}_{2}$ it doesn't form a geometric sequence.

So the sequence is an arithmetic sequence.