Problem 13 - Discrete Mathematics with Applications

Question

Ned deposits a certain amount \(A_{0}\) in a bank at an annual interest rate of \(12 \%\) compounded annually. The compound amount he would receive at the end of \(n\) years is given by \(A_{n}=1.12 \mathrm{A}_{n-1},\) where \(n \geq 1 .\) Determine the initial deposit \(A_{0}\) if he would receive: \(\$ 1804.64\) at the end of 5 years.

Step-by-Step Solution

Answer

The initial deposit \(A_0\) is approximately \(\$1024\).

Step 1: Express the amount after 5 years in terms of \(A_0\)

Using the given recursive formula, we can find the amount after each year in terms of the initial deposit: For \(n=1\), we have: \(A_1 = 1.12A_{0}\) For \(n=2\), we have: \(A_2 = 1.12A_{1} = 1.12(1.12A_{0}) = (1.12)^2A_{0}\) Similarly, for \(n=3, n=4\), and \(n=5\), we would get: \(A_3 = (1.12)^3A_{0}\) \(A_4 = (1.12)^4A_{0}\) \(A_5 = (1.12)^5A_{0}\)

Step 2: Plug in the given amount after 5 years and solve for \(A_0\)

We are given that the amount Ned receives after 5 years is \(\$1804.64\). We plug this into the equation for \(A_5\): \(1804.64 = (1.12)^5A_{0}\) To solve for \(A_0\), we divide both sides of the equation by \((1.12)^5\): \(A_{0} = \frac{1804.64}{(1.12)^5}\) Now, we can calculate the value of \(A_0\): \(A_{0} = \frac{1804.64}{1.76234} \approx 1024\) So the initial deposit \(A_0\) is approximately \(\$1024\).

Key Concepts

Recursive Formula

A recursive formula can be thought of as a way to express a sequence where each term is defined based on its preceding term(s). In the context of compound interest, the recursive formula describes how the amount of money grows over time.
For Ned's deposit, the formula is given by:

• The initial deposit, or starting amount, is denoted as \( A_0 \).
• For any year \( n \), where \( n \geq 1 \), the amount at the end of the year \( n \), \( A_n \), is calculated as \( A_n = 1.12 A_{n-1} \).

This formula indicates that each year, the previous year's amount is multiplied by 1.12, representing a 12% increase. Over time, this recursive multiplication reflects the effect of compounding interest, where each year's earned interest is added to the principal, and this cumulative amount earns interest in subsequent years. By building upon prior terms, such a formula not only reveals growth but also allows us to trace back to the initial investment amount given any future value.

Interest Rate

The interest rate is a critical factor in understanding compound interest, as it defines the percentage by which the initial deposit or investment grows over each compounding period.
In this scenario, Ned's bank offers an annual interest rate of 12%. This means that every year, his initially deposited amount increases by 12%.

• To express this increment mathematically, if \( A_0 \) is the initial amount, after one year, it becomes \( A_1 = 1.12 A_0 \).
• The increase from \( A_0 \) to \( A_1 \) directly results from the interest rate: 12% is translated to a decimal as 0.12, so the annual factor is 1 + 0.12 = 1.12.

Interest rates can significantly affect the growth of an investment over time. Higher rates lead to rapidly growing balances due to the compounding effect, as each year's interest calculation includes both the principal and previously accumulated interest. Understanding how interest rates influence growth helps one anticipate future values or required initial deposits depending on financial goals.

Initial Deposit Calculation

The calculation of the initial deposit, or the amount originally invested, plays a key role in compound interest-related problems. To find this amount, given a future value and a constant compound interest rate, involves a few straightforward steps using the recursive formula:

• First, identify the final amount received after a certain number of years, which in our example is \( \\(1804.64 \) after 5 years.
• Using the expression from the recursive formula for 5 years, \( A_5 = (1.12)^5 A_0 \), where \( (1.12)^5 \) calculates the growth factor over 5 years, equaling approximately 1.76234.
• To solve for the initial deposit \( A_0 \), rearrange the formula to \( A_0 = \frac{A_5}{(1.12)^5} \).
• Substituting \( A_5 = 1804.64 \), you calculate \( A_0 = \frac{1804.64}{1.76234} \), resulting in approximately \( \\)1024 \).

This method provides the initial amount required to achieve a target sum, considering the accumulation effects of compounded interest over the chosen time frame.