**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

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Answered: Prove the continuous growth formmula…

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Prove the continuous growth formmula using the com interest formala. Hinti use the defination of tthe no le as a limit nd aigebraic calculations.

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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Question

**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

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