1. **State the problem:** We want to find the initial investment amount $P$ that will grow to $2500$ in $5$ years with a continuous compounding interest rate of $6\%$ per year. 2. **Formula used:** The formula for continuous compounding is $$A = Pe^{rt}$$ where: - $A$ is the amount after time $t$, - $P$ is the initial principal (what we want to find), - $r$ is the annual interest rate (as a decimal), - $t$ is the time in years, - $e$ is Euler's number (approximately $2.71828$). 3. **Plug in known values:** $$2500 = P e^{0.06 \times 5}$$ 4. **Simplify the exponent:** $$2500 = P e^{0.3}$$ 5. **Solve for $P$:** $$P = \frac{2500}{e^{0.3}}$$ 6. **Show cancellation step:** $$P = \frac{2500}{\cancel{e^{0.3}}} \times \frac{\cancel{1}}{1}$$ 7. **Calculate $e^{0.3}$:** $$e^{0.3} \approx 1.349858807576003$$ 8. **Calculate $P$:** $$P = \frac{2500}{1.349858807576003} \approx 1852.32$$ **Final answer:** The initial investment should be approximately **1852.32** to have 2500 in 5 years with continuous compounding at 6%.