# 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. ## High School Math for Life: Making Sense of Earnings

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