1. Let's consider a common applied math problem: calculating the compound interest on an investment. 2. The problem: Suppose you invest a principal amount $P$ at an annual interest rate $r$ compounded $n$ times per year for $t$ years. What is the amount $A$ after $t$ years? 3. The formula for compound interest is: $$A = P \left(1 + \frac{r}{n}\right)^{nt}$$ 4. Important rules: - $P$ is the initial principal (the starting amount). - $r$ is the annual interest rate expressed as a decimal (e.g., 5% = 0.05). - $n$ is the number of times interest is compounded per year. - $t$ is the number of years. 5. Example: If you invest $1000$ at an annual interest rate of $5\%$ compounded quarterly ($n=4$) for $3$ years, then: $$A = 1000 \left(1 + \frac{0.05}{4}\right)^{4 \times 3} = 1000 \left(1 + 0.0125\right)^{12} = 1000 \times (1.0125)^{12}$$ 6. Calculate $(1.0125)^{12}$: $$ (1.0125)^{12} \approx 1.159274 $$ 7. Multiply by the principal: $$ A \approx 1000 \times 1.159274 = 1159.27 $$ 8. So, after 3 years, the investment grows to approximately $1159.27$. This shows how compound interest causes the investment to grow faster than simple interest due to interest on interest.