Problem 1 - Algebra 2 and Trigonometry

Question

Show that the formula \(A=A_{0}(1+r)^{n}\) is equivalent to \(A=A_{0}(2)^{n}\) when \(r=100 \%\)

Step-by-Step Solution

Answer

The formulas are equivalent when \( r = 100\% \).

Step 1: Understand the Given Formula

We begin with the formula \( A = A_{0}(1 + r)^{n} \). This is the formula for future value in compound interest, where \( A_{0} \) is the initial amount, \( r \) is the interest rate, and \( n \) is the number of periods.

Step 2: Identify the Specific Case

Identify that we need to prove equivalence for the case when the interest rate \( r \) is 100%. Therefore, replace \( r \) with 1 in the formula.

Step 3: Substitute and Simplify

Substitute \( r = 1 \) into the formula: \[A = A_{0}(1 + 1)^{n} = A_{0}(2)^{n}\]This simplifies to the formula \( A = A_{0} (2)^{n} \).

Step 4: Verify Equivalence

Notice that the simplified formula \( A = A_{0}(2)^{n} \) matches exactly what we were tasked to show. This confirms that the initial formula is equivalent to the given formula when \( r = 100\% \).

Key Concepts

Future Value

In the context of compound interest, the future value represents the amount of money that an initial investment, often called the principal or initial amount, will grow to after a specific period. This growth is governed by the application of interest over time. The standard formula to calculate the future value when compounded at a specific interest rate is given by:
\[ A = A_{0} (1 + r)^n \]Where:

• \(A\) is the future value of the investment.
• \(A_{0}\) is the initial amount or principal.
• \(r\) is the interest rate per period.
• \(n\) is the number of compounding periods.

Understanding future value helps in evaluating how an investment will appreciate over time, making it a crucial concept in finance and savings planning.

Interest Rate

The interest rate in the compound interest formula is a critical factor in determining how much your investment grows. It is usually expressed as a percentage and represents the cost of borrowing money or the return on investment when saving money. In our formula:
\[ r \]is the interest rate per compounding period.### Types of Interest Rates

• Fixed Interest Rate: The rate does not change over the duration of the investment.
• Variable Interest Rate: The rate may fluctuate, often depending on market conditions.

In our specific example, when the interest rate is 100%, it highlights a unique scenario where the principal amount doubles during each period. This doubling directly influences the equation, making the interest rate crucial in calculating future growth.

Exponential Growth

Exponential growth is a key principle behind compound interest, where quantities grow at a consistent rate over time, producing a curve that becomes steeper as it progresses. This type of growth is characterized by the formula:
\[ A = A_{0} (1 + r)^n \]As you compound more often or increase the rate, the power of exponentials truly shines; the investment grows at an increasingly rapid pace. This effect is particularly evident in our exercise, where:### Example

• The interest rate \( r \) is set to 100%, doubling the principal after every period.
• This converts the base of the exponential function from \(1 + 1 = 2\), simplifying the scenario to \(A = A_{0} (2)^n\).

This form of exponential growth illustrates how quickly values can increase with compound interest, making it a powerful tool for wealth accumulation over time.

Proof Technique

A proof technique in mathematics involves logically showing that a given statement or formula is true. In our exercise, the goal was to prove that the compound interest formula with a 100% interest rate yields a doubled initial value each period:\[ A = A_{0} (1 + 1)^n = A_{0} (2)^n \]The steps to prove equivalence included:

• Starting from the known formula \(A = A_{0} (1 + r)^n\).
• Substituting \(r = 1\) for the 100% rate.
• Simplifying to reach the expression \(A = A_{0} (2)^n\).
• Verifying and confirming that both expressions represent the same situation under the specified condition.

By proving the equivalence, we see how using such techniques in mathematics ensures the logical consistency and correctness of formulas under varying conditions. This not only strengthens understanding but also broadens the applicability of mathematical concepts in real-world scenarios.