Problem 67 - Algebra and Trigonometry

Question

Compound Interest \(\quad\) A savings account earns 5\(\%\) interest compounded annually. If you invest \(x\) dollars in such an ac- count, then the amount \(A(x)\) of the investment after one year is the initial investment plus 5\(\%\) , that is, $$ A(x)=x+0.05 x=1.05 x $$ Find $$ \begin{array}{l}{A \circ A} \\ {A \circ A \circ A} \\ {A \circ A \circ A \circ A}\end{array} $$ What do these compositions represent? Find a formula for what you get when you compose \(n\) copies of \(A\) .

Step-by-Step Solution

Answer

\(A^n(x) = 1.05^n x\); represents the amount after \(n\) years.

Step 1: Understanding the Given Function

The function for the amount after one year is given by \(A(x) = 1.05x\). This tells us that each time we apply the function \(A\), we multiply our initial amount by 1.05 to account for the 5% interest compounded yearly.

Step 2: Calculating \(A \circ A\)

To find \(A \circ A\), we want to apply the function \(A\) twice. This is calculated as: \(A(A(x)) = A(1.05x) = 1.05(1.05x) = 1.05^2x\). This represents the amount after two years.

Step 3: Calculating \(A \circ A \circ A\)

To find \(A \circ A \circ A\), we apply \(A\) three times: \(A(A(A(x))) = A(1.05^2x) = 1.05(1.05^2x) = 1.05^3x\). This represents the amount after three years.

Step 4: Calculating \(A \circ A \circ A \circ A\)

Here, we apply \(A\) four times: \(A(A(A(A(x)))) = A(1.05^3x) = 1.05(1.05^3x) = 1.05^4x\). This represents the amount after four years.

Step 5: General Formula for \(n\) Years

After observing the pattern, the general composition \(A\) applied \(n\) times is \(A^{n}(x) = 1.05^nx\). This represents the compounded amount after \(n\) years.

Key Concepts

Interest Rate

The interest rate is an essential concept in understanding how money grows over time in a savings account. In the example provided, the interest rate is 5% annually. When you have money in a savings account or any investment, an interest rate determines how much your initial deposit, referred to as the principal, grows over each period. For our exercise:

- The interest rate is expressed as a percentage (5%).
- It determines the additional amount added to your initial investment each year.

Calculating the interest: If you invest an amount 'x', a 5% interest translates to 0.05 times 'x'. This means after one year, your investment grows to 1.05 times the original amount. Understanding the interest rate is critical to predicting how your investments will perform over time.

Annual Compounding

Annual compounding is a specific method of calculating interest in which the interest earned over a year is added to the principal at the end of each year. Then, the new total becomes the principal for the next year. In the context of our example:

- Your initial investment is multiplied by 1.05 at the end of each year.
- The newly calculated total forms the basis of next year's interest calculation.

Think of annual compounding as a snowball effect; the interest earned each year gets rolled into the investment, leading to potentially exponential growth in your total investment value when the interest is applied year after year. It differs from simple interest, where only the principal earns interest every year.

Composed Functions

Composed functions are a mathematical concept where one function is applied to the result of another function. In financial mathematics, they are handy for calculating how investments grow over multiple periods when compounding is involved. In our given exercise:

- The function for one year is noted as \( A(x) = 1.05x \).
- Calculating \( A \circ A \) involves applying this function twice, leading to \( A(A(x)) = 1.05^2x \), which signifies growth over two years.

By repeatedly applying the function \( A \) to itself, you effectively forecast how investment grows over multiple years. Each additional composition represents extending your timeline by another year and is a powerful tool for financial planning.

Exponential Growth

Exponential growth describes a process where the quantity increases at a consistent rate relative to its current value. With compound interest, this is a natural outcome, as interest is continually added to the increasing balance. This is best captured by:

- The formula \( A^n(x) = 1.05^nx \), where \( n \) is the number of years.
- As years pass, the growth compounding resembles exponential growth due to its geometrically increasing pattern.

In essence, compounding turns linear growth into exponential growth. This means your investment doesn't just grow steadily but rather accelerates, becoming faster with each period, leading to significant returns as time progresses. Exponential growth is what makes saving with compound interest a very effective strategy over a long period.