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

Question

Phoenix Morse · Accepted Answer

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:

d_{1} = a_{2} - a_{1} = - 5 - (15) = - 20

d_{2} = a_{3} - a_{2} = - 25 - (- 5) = - 20

d_{3} = a_{4} - a_{3} = - 45 - (- 25) = - 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:

r_{1} = \frac{a_{2}}{a_{1}} = - \frac{5}{15} = - \frac{1}{3}

r_{2} = \frac{a_{3}}{a_{2}} = \frac{- 25}{- 5} = 5

Since

r_{1}

is not equal to

r_{2}

it doesn't form a geometric sequence.

So the sequence is an arithmetic sequence.

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

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