Problem 66 - Calculus: Early Transcendentals

Question

If you deposit \( \$ \)100 at the end of every month into an account that pays \( 3 \% \) interest per year compounded monthly, the amount of interest accumulated after \( n \) months is given by the sequence \( I_n = 100 \left( \frac {1.0025^n - 1}{0.0025} - n\right) \) (a) Find the first six terms of the sequence. (b) How much interest will you have earned after two years?

Step-by-Step Solution

Answer

First six terms: 0, 0.25, 0.752, 1.524, 2.528, 3.784. After two years: $73.416.

Step 1: Understanding the formula

The given formula for the interest accumulated after \( n \) months is \( I_n = 100 \left( \frac {1.0025^n - 1}{0.0025} - n\right) \). This is a sequence where we need to find specific terms by substituting values of \( n \).

Step 2: Calculate the first term when n=1

Substitute \( n = 1 \) into the formula: \[ I_1 = 100 \left( \frac {1.0025^1 - 1}{0.0025} - 1 \right) \]. Calculate each part: \( 1.0025^1 = 1.0025 \), and \( \frac{1.0025 - 1}{0.0025} = 1 \), so \( I_1 = 100 \times (1 - 1) = 0 \).

Step 3: Calculate the second term when n=2

Substitute \( n = 2 \) into the formula: \[ I_2 = 100 \left( \frac{1.0025^2 - 1}{0.0025} - 2 \right) \]. Calculate: \( 1.0025^2 \approx 1.00500625 \), and \( \frac{1.00500625 - 1}{0.0025} \approx 2.0025 \), so \( I_2 = 100 \times (2.0025 - 2) = 0.25 \).

Step 4: Calculate the third term when n=3

Substitute \( n = 3 \) into the formula: \[ I_3 = 100 \left( \frac{1.0025^3 - 1}{0.0025} - 3 \right) \]. Calculate: \( 1.0025^3 \approx 1.0075188 \), and \( \frac{1.0075188 - 1}{0.0025} \approx 3.00752 \), so \( I_3 = 100 \times (3.00752 - 3) = 0.752 \).

Step 5: Calculate the fourth term when n=4

Substitute \( n = 4 \) into the formula: \[ I_4 = 100 \left( \frac{1.0025^4 - 1}{0.0025} - 4 \right) \]. Calculate: \( 1.0025^4 \approx 1.0100381 \), and \( \frac{1.0100381 - 1}{0.0025} \approx 4.01524 \), so \( I_4 = 100 \times (4.01524 - 4) = 1.524 \).

Step 6: Calculate the fifth term when n=5

Substitute \( n = 5 \) into the formula: \[ I_5 = 100 \left( \frac{1.0025^5 - 1}{0.0025} - 5 \right) \]. Calculate: \( 1.0025^5 \approx 1.0125632 \), and \( \frac{1.0125632 - 1}{0.0025} \approx 5.02528 \), so \( I_5 = 100 \times (5.02528 - 5) = 2.528 \).

Step 7: Calculate the sixth term when n=6

Substitute \( n = 6 \) into the formula: \[ I_6 = 100 \left( \frac{1.0025^6 - 1}{0.0025} - 6 \right) \]. Calculate: \( 1.0025^6 \approx 1.0150946 \), and \( \frac{1.0150946 - 1}{0.0025} \approx 6.03784 \), so \( I_6 = 100 \times (6.03784 - 6) = 3.784 \).

Step 8: Determine interest after two years (n=24)

After two years, \( n = 24 \) months. Substitute into the formula: \[ I_{24} = 100 \left( \frac{1.0025^{24} - 1}{0.0025} - 24 \right) \]. Calculate: \( 1.0025^{24} \approx 1.0618354 \), and \( \frac{1.0618354 - 1}{0.0025} \approx 24.73416 \), so \( I_{24} = 100 \times (24.73416 - 24) = 73.416 \).

Key Concepts

Mathematical Sequences

Mathematical sequences are an essential tool in understanding complex financial scenarios. Sequences are lists of numbers arranged in a specific order. In financial mathematics, they often help us calculate things like accumulated interest over time. They provide a step-by-step guide to determining how amounts grow or decrease.

• Each number in a sequence is known as a 'term'.
• The sequence follows a particular pattern or formula.

In our exercise, the sequence is created using the formula \( I_n = 100 \left( \frac{1.0025^n - 1}{0.0025} - n \right) \) to find how much interest has accumulated after \( n \) months.
This sequence can be seen as a type of arithmetic sequence embedded into a compound interest formula. Calculating terms individually can help you visualize and understand how the interest grows month by month.For instance, when substituting \( n = 1, 2, 3, \)... into the formula, you systematically see how the interest amount increases as you go from month to month. This gradual understanding is key to grasping how sequences underpin financial computations.

Interest Calculation

Interest calculation, particularly compound interest, is pivotal in financial mathematics. It refers to the interest on a principal amount and the interest that it has already accumulated. This is unlike simple interest, which is calculated only on the principal amount.Our exercise uses a formula to illustrate how compound interest grows over time:

• The annual interest rate is divided monthly, giving a smaller percentage each month.
• This monthly interest accumulates not just on the initial principal, but also on the interest already gained.

For our question, it's important to note that the interest rate is 3% per year, which converts to 0.25% per month. That's calculated using the expression \(1.0025^n\), where 1.0025 is the growth factor per month.
As you continue substitution, each new term represents a new calculation of interest that considers the growth of previous months' interest. Understanding each term in this sequence approach helps you visualize how compound interest is powerful over time.

Financial Mathematics

Financial mathematics is a branch of applied mathematics that assesses monetary risk and helps with investment decisions through tools like compound interest. It empowers individuals and businesses to make informed financial choices by understanding the principles of money growth and loss over time. Utilizing formulas and sequences can:

• Help you understand the financial market and lender agreements better.
• Enable accurate predictions for investment growth, as demonstrated in the exercise.

By the end of two years, the consistency of monthly deposits and the compounding effects result in total interest of 73.416. This demonstrates how small, regular investments can generate significant returns over time.
In everyday applications, financial mathematics and its principles are incredibly useful to optimize returns and manage loans, investments, or debts effectively. Mastering such concepts better prepares you for making sound financial decisions.