Question

Sebastian writes the recursive formula f(x+1)=4f(x) to represent a geometric sequence whose second term is 12.

Which explicit formula can be used to model the same sequence?

A. F(x)=12(4)x
B. F(x)=3(4)^x-1
C. F(x)=4(12)^x
D. F(x)=4(3)^x-1

Answer 1

b., each time you multiply by 4 so you are increasing by 4, exponentially, you can also literally eliminate all the other choices 

Answer 2

B. f(x)=3(4)^x-1

Explanation:

In order to know which explicit formula can be used to model the same sequence you have to calculate f(x) in:

f(x+1) and 4f(x)

and prove if they have the same results in both sides (f(x+1) = 4f(x))

Then, if you take
`f(x)=3(4)^(x-1)`

1. f(x+1)

`f(x+1)=3(4)^((x+1)-1)`=
`f(x+1)=3(4)^(x)`

2. 4f(x)

`4f(x)`=
`4*3(4)^(x-1)`=
`12(4)^(x-1)`

if you evaluate both sides (f(x+1) and 4f(x)) for x=0, x=1, x=2, you have:

x=0

`f(x+1)=3(4)^(x+1)-1`=
`3(4)^0`=3

`4f(x)`=
`4*3(4)^(x-1)`=
`12(4)^-1`=3

x=1

`f(x+1)=3(4)^((x+1)-1)`=
`3(4)^1`=12

`4f(x)`=
`4*3(4)^(x-1)`=
`12(4)^0`=12

x=2

`f(x+1)=3(4)^((x+1)-1)`=
`3(4)^2`=48

`4f(x)`=
`4*3(4)^(x-1)`=
`12(4)^1`=48

Then you can say for: f(x)=3(4)^x-1 is fulfilled that f(x+1) = 4f(x)