Problem 10 - Discrete Mathematics with Applications

Question

A person deposits \(\$ 1000\) in a bank at an annual interest rate of \(6 \% .\) Let \(A(n)\) denote the compound amount she will receive at the end of \(n\) interest periods. Define \(A(n)\) recursively if interest is compounded: Semiannually

Step-by-Step Solution

Answer

The recursive formula for the compound amount after n periods of semi-annual compounding can be defined as: Base case: \[A(0) = 1000\] Recursive formula: \[A(n) = A(n-1) \times \left(1 + \frac{0.06}{2}\right), \text{ for } n \geq 1\]

Step 1: Understanding compound interest

Compound interest is the interest earned not only on the initial deposit but also on any accumulated interest from previous periods. In this case, the interest is compounded semi-annually, which means it is calculated and added to the principal twice a year. To find the compound amount at the end of any period, we use the formula: \[A(n) = P(1 + r/n)^{nt}\] Where: - A(n): compound amount after n periods - P: principal amount or the initial deposit - r: annual interest rate (given as a decimal) - n: number of times interest is compounded per year - t: number of years

Step 2: Converting the interest rate to a decimal

We are given the annual interest rate as 6%. We need to convert this percentage into a decimal form before using it in the formula. To do this, we divide the percentage by 100: \[r = \frac{6}{100}= 0.06\]

Step 3: Applying semi-annual compounding

Since the interest is compounded semi-annually, the number of compounding periods in a year, n, is 2. Therefore, the interest rate per period will be \(r/2\). With this information, we can now create a recursive formula for the compound amount A(n) after n interest periods.

Step 4: Creating a recursive formula for A(n)

The base case is A(0) since n starts at 0 when there are no compounding periods. At this point, the compound amount is equal to the initial deposit: \[A(0) = 1000\] Next, we want to find a general formula for A(n) based on the previous term A(n-1). Since the interest is added twice per year (semi-annually), we will use the following formula to get the compound amount for the next period: \[A(n) = A(n-1) \times \left(1 + \frac{r}{2}\right)\] So the recursive formula for the compound amount after n periods of semi-annual compounding is: Base case: \[A(0) = 1000\] Recursive formula: \[A(n) = A(n-1) \times \left(1 + \frac{0.06}{2}\right), \text{ for } n \geq 1\]

Key Concepts

Compound Interest

Compound interest is an amazing way to grow your savings over time. Unlike simple interest, which only calculates interest on the original amount deposited (the principal), compound interest also factors in interest earned from previous periods. Essentially, it allows you to earn "interest on interest," which can cause your money to grow at an accelerated rate.

Understanding how compound interest works involves using a specific formula:

• \[A(n) = P(1 + \frac{r}{n})^{nt}\]
• \(A(n)\) represents the compound amount after a certain number of periods.
• \(P\) is the principal or initial deposit.
• \(r\) signifies the annual interest rate, expressed as a decimal.
• \(n\) is the number of compounding periods within one year.
• \(t\) refers to the number of years the money is invested or borrowed for.

By using this formula, you can determine how much your initial investment will grow over each compounding period, making compound interest a powerful tool for wealth growth.

Semi-Annual Compounding

Semi-annual compounding is a specific method of calculating compounded interest. This technique involves applying the interest rate to your balance twice a year, rather than annually. By doing this, the process effectively divides the annual interest rate into two, reflecting the two compounding periods each year.

To incorporate semi-annual compounding into the formula for compound interest, the annual interest rate \(r\) is divided by 2, as there are two compounding periods in a year. For example, with an annual interest rate of 6\%, the semi-annual interest rate becomes \(\frac{6}{2} = 3\%\) or \(0.03\) in decimal form.

• Semi-annual means two compounding periods per year.
• The interest rate per compounding period is half of the annual rate.

By understanding how semi-annual compounding affects the growth of your investment, you can better predict and plan for future increases in your compound amount, making more strategic financial decisions.

Interest Rate

The interest rate is a critical component for any investment or loan as it determines how much you will earn or pay over time. It's typically presented as a percentage on an annual basis and can come in different types, such as simple or compound.

When working with compound interest, it's essential to convert the given annual percentage rate into a decimal before using it in calculations. For example, an interest rate of 6\% becomes \(0.06\) when expressed in decimal form.

The frequency of compounding periods also affects how the interest rate is applied. For semi-annual compounding, you need to adjust the annual interest rate by dividing it by the number of compounding periods in a year, effectively creating a per-period rate:

• Annual rate conversion: \(\text{6\%} = 0.06\)
• Per-period rate for semi-annual: \(\frac{0.06}{2} = 0.03\)

By grasping how interest rates work and how they are calculated for different compounding frequencies, you can accurately forecast and assess how your investments will grow over time.