Question

Given that a geometric sequence can be expressed as a non-linear function in the form f(x) = abx, write a non-linear function to describe the sequence {3, 6, 12, 24, 48, 96, ...}.

Answer 1

Answer:

$f(x)=\dfrac{3}{2}(2)^x$

Step-by-step explanation:

The given sequence is

{3, 6, 12, 24, 48, 96, ...}

It means f(x)=3, 6, 12, 24, 48, 96, ... for x=1,2,3,4,5,6,.. respectively.

We need to find non-linear function in the form  

$f(x)=ab^x$       ...(1)

We know that f(x)=3 for x=1.

$3=ab^1$          ....(i)

We know that f(x)=6 for x=2.

$6=ab^2$          ....(ii)

Divide equation (ii) by (i).

$\dfrac{6}{3}=\dfrac{ab^2}{ab}$

$2=b$

Substitute b=2 in equation (1).

$3=a(2)^1$

$\dfrac{3}{2}=a$

Substitute $a=\dfrac{3}{2}$ and $b=2$ in (1).

$f(x)=\dfrac{3}{2}(2)^x$

Therefore, the required function is $f(x)=\dfrac{3}{2}(2)^x$.

Answer 2

$f(x) = 3 \times {2}^{x}$