Problem 72 - Precalculus Mathematics for Calculus

Question

Compound Interest Helen deposits \(\$ 100\) at the end of each month into an account that pays \(6 \%\) interest per year compounded monthly. The amount of interest she has accumulated after \(n\) months is given by the sequence $$ I_{n}=100\left(\frac{1.005^{n}-1}{0.005}-n\right) $$ (a) Find the first six terms of the sequence. (b) Find the interest she has accumulated after 5 years.

Step-by-Step Solution

Answer

First six terms: 0, 0.5025, 1.5075, 2.0106, 3.0256, 2.5377. After 5 years: 977.53 in interest.

Step 1: Understanding the Formula

The given formula for accumulated interest is \( I_n = 100\left(\frac{1.005^n - 1}{0.005} - n\right) \). Here, \( n \) represents the number of months. We need to calculate this expression for each term to find the interest accumulated each month.

Step 2: Calculate the First Term \( I_1 \)

For \( n = 1 \), substitute into the formula: \[ I_1 = 100\left(\frac{1.005^1 - 1}{0.005} - 1\right) \]This simplifies to:\[ I_1 = 100\left(\frac{0.005}{0.005} - 1\right) = 100(1 - 1) = 0 \]

Step 3: Calculate the Second Term \( I_2 \)

For \( n = 2 \), substitute into the formula: \[ I_2 = 100\left(\frac{1.005^2 - 1}{0.005} - 2\right) \]This simplifies to:\[ I_2 = 100\left(\frac{1.010025 - 1}{0.005} - 2\right) = 100(2.005 - 2) = 0.5025 \]

Step 4: Calculate Remaining Terms \( I_3 \) to \( I_6 \)

Apply the same calculation method for \( n = 3 \) to \( n = 6 \):- \( I_3 = 100\left(\frac{1.005^3 - 1}{0.005} - 3\right) = 1.5075125 \)- \( I_4 = 100\left(\frac{1.005^4 - 1}{0.005} - 4\right) = 2.01055063 \)- \( I_5 = 100\left(\frac{1.005^5 - 1}{0.005} - 5\right) = 3.02560376 \)- \( I_6 = 100\left(\frac{1.005^6 - 1}{0.005} - 6\right) = 2.53768139 \)

Step 5: Conclusion for Part (a)

The first six terms of the sequence are: 1. \( I_1 = 0 \)2. \( I_2 = 0.5025 \)3. \( I_3 = 1.5075125 \)4. \( I_4 = 2.01055063 \)5. \( I_5 = 3.02560376 \)6. \( I_6 = 2.53768139 \)

Step 6: Calculate Interest After 5 Years

First, determine the number of months in 5 years: \( 5 \times 12 = 60 \). Now substitute \( n = 60 \) into the formula:\[ I_{60} = 100\left(\frac{1.005^{60} - 1}{0.005} - 60\right) \] This simplifies to:\[ I_{60} = 100\left(\frac{1.34885 - 1}{0.005} - 60\right) = 100\times(69.77 - 60) = 977.53 \]

Step 7: Conclusion for Part (b)

After 5 years (60 months), Helen has accumulated approximately \( 977.53 \) in interest.

Key Concepts

interest calculation

Interest calculation is a fundamental concept in financial mathematics used to determine the reward for investing money over a period of time. In the context of compound interest, this calculation considers not just the initial principal but also the interest that has been added to it over time. For example, in Helen's case, her interest calculation uses the formula: \[ I_n = 100\left(\frac{1.005^n - 1}{0.005} - n\right) \] This formula allows us to determine how much interest she has accumulated after depositing money monthly into an account that compounds interest. Every month, the account adds interest to both the initial deposit and any previously earned interest, leading to potential exponential growth of Helen's savings. Key points to remember about interest calculation:

• Interest is calculated on the total amount of money at regular intervals, not just the initial investment.
• In compound interest, each time interest is calculated, it is based on the current total, which includes previous interest earnings.
• The frequency of compounding (monthly, in this case) greatly affects the growth of investment.

sequences in mathematics

Sequences in mathematics often represent ordered lists of numbers that follow a certain pattern or rule. In problems involving compound interest, sequences can describe the monthly accrual of interest. With Helen's account, the interest accumulation is expressed as a sequence given by the formula for \( I_n \). Each term of this sequence represents the interest accrued by the end of a specific month. When analyzing sequences:

• The rule or formula for the sequence is crucial as it determines each term's value.
• Examining the sequence's behavior over time can provide insights into how investments perform.
• Different types of sequences (e.g., arithmetic or geometric) have distinct patterns and applications.

For Helen, each resulting term \( I_1, I_2, \ldots, I_6 \) gives us the specific interest gained by following the formula, illustrating how the investment grows month by month.

financial mathematics

Financial mathematics involves using mathematical models and formulas to solve problems related to finance. This field is essential for making informed decisions about investments, loans, savings, and much more. In Helen's case, financial mathematics offers the tools needed to calculate compound interest accurately and understand the future value of an investment. Key elements of financial mathematics include:

• Understanding different types of interest - simple versus compound.
• Using formulas effectively to model real-world financial situations, such as the accumulation of interest over time.
• Applying mathematical techniques to evaluate different saving and investment options.

Through financial mathematics, individuals like Helen can optimize their savings strategies, predict the growth of their investments, and make better financial decisions. These mathematical insights are vital for both personal finance management and larger economic considerations.