Question

What is the common ratio and explicit formula for -2,6,-18,54?

Answer 1

The explicit formula for a given geometric sequence is of the form

`a_(4) =(1)/((a_(1))^4)`

Explanation:

Given sequence is
`{\{-2,6,-18,54}\}`

Let
`a_(1)=-2,a_(2)=6,a_(3)=-18,a_(4)=54`

To find the common ratio r:

`r=(a_(2))/(a_(1))`

`=(6)/(-2)`

`=-3`

Therefore r=-3

`r=(a_(3))/(a_(2))`

`=(-18)/(6)`

`=-3`

Therefore r=-3

Therefore common ratio is -3

Therefore given sequence is geometric sequence.

The explicit formula for a geometric sequence is
`a_(n)=a_(1)^(r-1)`, where common ratio is r.

From given sequence we are having r=-3

Therefore the explicit formula for a given geometric sequence is of the form
`a_(4) =a_(1)^(-3-1)`

`a_(4) =a_(1)^(-4)`

`a_(4) =(1)/((a_(1))^4)`