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: \(f(0)=162\)

\(\displaystyle{f{{\left({n}\right)}}}=\frac{{1}}{{3}}\cdot{f{{\left({n}-{1}\right)}}}\)