Exponential Decay
Consider a radioactive substance with an original weight $30$ grams. It $\textit{decays}$ or reduces by $4\%$ each year. The multiplier for this is $96\%$ or $0.96$.
When the multiplier is less than $1$, we call it $\textit{Exponential Decay}$.
If $R_n$ is the weight after $n$ years, then:
\( \begin{align} \displaystyle
R_0 &= 30 \\
R_1 &= R_0 \times 0.96 = 30 \times 0.96 = 28.8 \\
R_2 &= R_1 \times 0.96 = 30 \times 0.96 \times 0.96 = 30 \times 0.96^2 \approx 27.6 \\
R_3 &= R_2 \times 0.96 = 30 \times 0.96^2 \times 0.96 = 30 \times 0.96^3 \approx 26.5 \\
\vdots \\
R_{20} &= 30 \times 0.96^{20} \approx 13.3 \\
\vdots \\
R_{50} &= 30 \times 0.96^{50} \approx 3.90 \\
\vdots \\
R_{100} &= 30 \times 0.96^{100} \approx 0.51
\end{align} \)
and from this pattern, we see that $R_n = 0.96^n$.
Example 1
Determine whether $Y_n = 0.8^n$ is exponential decay.
$0.8<1$
Yes, $Y_n$ undergoes exponential decay.
Example 2
Determine whether $P_n = 2.3^{-0.7n}$ is exponential decay.
\( \begin{align} \displaystyle
P_n &= 2.3^{-0.7n} \\
&= (2.3^{-0.7})^n \\
&= 0.558 \cdots^n \\
\end{align} \)
$0.558 \cdots <1$
Thus $P_n$ undergoes exponential decay.
Example 3
Determine whether $J_n = 0.8^{-0.5n}$ is exponential decay.
\( \begin{align} \displaystyle
J_n &= 0.8^{-0.5n} \\
&= (0.8^{-0.5})^n \\
&= 1.12 \cdots^n \\
\end{align} \)
$1.12>1$
Thus $J_n$ does not undergo exponential decay.
Example 4
Determine whether $N_n =200 \times 0.95^n$ is exponential decay.
$0.95<1$
Thus $N_n$ undergoes exponential decay.
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