1. The problem is to understand the compound interest formula: $$A = P \left(1 + \frac{r}{n}\right)^{nt}$$ which calculates the amount $A$ after $t$ years. 2. Here, $P$ is the principal (initial amount), $r$ is the annual interest rate (as a decimal), $n$ is the number of times interest is compounded per year, and $t$ is the time in years. 3. The formula works by adding the interest rate per compounding period $\frac{r}{n}$ to 1, then raising this to the power of the total number of compounding periods $nt$. 4. This exponentiation accounts for interest being compounded multiple times, which means interest is earned on previously earned interest. 5. For example, if $P=1000$, $r=0.05$, $n=4$, and $t=3$, 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. Calculating $1.0125^{12}$ gives approximately $1.1616$, so: $$A \approx 1000 \times 1.1616 = 1161.6$$ 7. This means after 3 years, the balance grows to approximately 1161.6. This formula is fundamental in finance for calculating compound interest growth over time.