Question

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

Answer 1

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.