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
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asked 2021-07-01

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

Answers (1)

2021-07-02

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: \(f(0)=162\)
\(\displaystyle{f{{\left({n}\right)}}}=\frac{{1}}{{3}}\cdot{f{{\left({n}-{1}\right)}}}\)

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