Question

# Show that an exponential model fits the data. Then write a recursive rule that models the data.n 0 1 2 3 4 5 f(n)162 54 18 6 2 2/3 ​

Exponential models

Show that an exponential model fits the data. Then write a recursive rule that models the data.
$$\begin{array}{|c|c|}\hline n & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline f(n) & 162 & 54 & 18 & 6 & 2 & \frac{2}{3} \\ \hline \end{array}$$

The given function represents an exponential model if every consecutive value is the previus value multiplied by a costant b that is called the growth (decay) factor. We not then that this table represents an exponential model, because the decay factors is $$\frac{1}{3}$$.
The recursive rule is $$f(0)=a$$ and $$\displaystyle{f{{\left({n}\right)}}}={b}\cdot{f{{\left({n}-{1}\right)}}}$$ with a the initial value (at $$n=0$$) and b is the growth factor (or decay factor if $$b<0$$).
Since $$a=162$$ and $$\displaystyle{b}=\frac{{1}}{{3}}$$ we then obtain the recursive rule: $$f(0)=162$$
$$\displaystyle{f{{\left({n}\right)}}}=\frac{{1}}{{3}}\cdot{f{{\left({n}-{1}\right)}}}$$