Question

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Which formula can be used to describe the sequence? Negative two-thirds, −4, −24, −144,... f(x) = 6(negative two-thirds) Superscript x minus 1 f(x) = −6(Two-thirds) Superscript x minus 1 f(x) = Negative two-thirds(6)x − 1 f(x) = Two-thirds(−6)x − 1

Answer 1

a

Explanation:

Answer 2

For the sequence is
`-(2)/(3)` ,-4 ,-24 ,-144 ,...

Hence the formula
`f(x)=-(2)/(3)(6)^(x-1)` for x=1,2,3,... represents the given geometric sequence

Explanation:

Given sequence is
`-(2)/(3)` ,-4 ,-24 ,-144 ,...

To find the formula to describe the given sequence :

Let
`a_1=(-2)/(3)` ,
`a_2=-4` ,
`a_3=-24`,...

First find the common ratio

`r=(a_2)/(a_1)` here
`a_1=(-2)/(3)` and,
`a_2=-4`

`=(-4)/((-2)/(3))`

`=(4* 3)/(2)`

`=(12)/(2)`

`r=6`

`r=(a_3)/(a_2)` here
`a_2=-4` and
`a_3=-24`

`=(-24)/(-4)`

`=6`

`r=6`

Therefore the common ratio is 6

Therefore the given sequence is geometric sequence

The nth term of the geometric sequence is

`a_n=a_1r^(n-1)`

The formula which describes the given geometric sequence is

`f(x)=a_1r^(x-1)` for x=1,2,3,...

`=(-2)/(3)6^(x-1)` for x=1,2,3,...

Now verify that
`f(x)=a_1r^(x-1)` for x=1,2,3,... represents the given geometric sequence or not

put x=1 and the value of
`a_1` in
`f(x)=a_1r^(x-1)` for x=1,2,3,...

we get
`f(1)=-(2)/(3)(6)^(1-1)`

`=-(2)/(3)(6)^0`

`=-(2)/(3)`

Therefore
`f(1)=-(2)/(3)`

put x=2 we get
`f(2)=-(2)/(3)(6)^(2-1)`

`=-(2)/(3)(6)^1`

`=-(12)/(3)`

Therefore
`f(2)=-4`

put x=3 we get
`f(3)=-(2)/(3)(6)^(3-1)`

`=-(2)/(3)(6)^2`

`=-(2(36))/(3)`

Therefore
`f(3)=-24`

Therefore the sequence is f(1),f(2),f(3),...

Therefore the sequence is
`-(2)/(3)` ,-4 ,-24 ,-144 ,...

Hence the formula
`f(x)=a_1r^(x-1)` for x=1,2,3,... represents the given geometric sequence is verified

Therefore the formula
`f(x)=-(2)/(3)(6)^(x-1)` for x=1,2,3,... represents the given geometric sequence