Answer:
B. f(x)=3(4)^x-1
Step-by-step 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)
