**Proving the Continuous Growth Formula using the Compound Interest Formula: A Step-by-Step Guide****Objective:**To understand and establish the continuous growth formula by deriving it from the compound interest formula using limits and algebraic calculations.**Introduction:**Continuous growth is a fundamental concept in various fields such as finance, biology, and economics. Here, we will derive the continuous growth formula, commonly expressed as \( A = Pe^{rt} \), from the compound interest formula.**Compound Interest Formula:**The compound interest formula is usually expressed as:\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]where:- \(A\) is the amount of money accumulated after n years, including interest.- \(P\) is the principal amount (the initial amount of money).- \(r\) is the annual interest rate (decimal).- \(n\) is the number of times that interest is compounded per year.- \(t\) is the time the money is invested for in years.**Derivation:**1. **Substitute the Compound Interest Equation:** We start from the standard compound interest equation: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]2. **Limit Definition:** Considering the limit as the number of compounding periods per year \( n \) approaches infinity, we get the definition of continuous growth. This transformation can be depicted as: \[ \lim_{n \to \infty} \left(1 + \frac{r}{n}\right)^{nt} \]3. **Applying Limits and Exponential Transformations:** To solve this limit, recognize it as a known exponential limit, leading to the natural number \( e \): \[ \lim_{n \to \infty} \left(1 + \frac{r}{n}\right)^n = e^r \]4. **Continuous Compound Interest Formula:** Therefore, as \( n \) tends to infinity: \[ A = P \left(\lim_{n \to \infty} \left(1 + \frac{r}{n}\right)^{nt}\right) = P e^{rt} \]**Conclusion:**Hence, we have derived the continuous growth formula:\[ A = Pe^{rt

Name: **Proving the Continuous Growth Formula using the Compound Interest F
Uploaded: 2024-03-20T06:38:58.000Z
Channel: Bartleby
Description: **Proving the Continuous Growth Formula using the Compound Interest Formula: A Step-by-Step Guide****Objective:**To understand and establish the continuous growth formula by deriving it from the compound interest formula using limits and algebraic c