Problem 70 - Precalculus: Mathematical for Calculus

Question

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 are 0, 0.5, 1.5, 3, 5.07, 7.58. After 5 years, the accumulated interest is 977.

Step 1: Understanding the Problem

We are given an interest sequence formula \(I_{n}=100\left(\frac{1.005^{n}-1}{0.005}-n\right)\). We need to find the first six terms of this sequence and calculate the interest accumulated after 5 years (60 months). This involves plugging different values of \(n\) into the formula and computing the result.

Step 2: Calculate the First Term (\(n=1\))

Substitute \(n = 1\) into the sequence formula. \[I_1 = 100\left(\frac{1.005^1-1}{0.005} - 1\right)\]Calculate the expression:\(1.005^1 = 1.005\)\(\frac{1.005 - 1}{0.005} = 1\)Thus:\[I_1 = 100 \times (1 - 1) = 0\]

Step 3: Calculate the Second Term (\(n=2\))

Substitute \(n = 2\) into the formula. \[I_2 = 100\left(\frac{1.005^2-1}{0.005} - 2\right)\]Calculate the expression:\(1.005^2 \approx 1.010025\)\(\frac{1.010025 - 1}{0.005} \approx 2.005\)Thus:\[I_2 = 100 \times (2.005 - 2) = 0.5\]

Step 4: Calculate the Third Term (\(n=3\))

Substitute \(n = 3\) into the formula. \[I_3 = 100\left(\frac{1.005^3-1}{0.005} - 3\right)\]Calculate the expression:\(1.005^3 \approx 1.015075125\)\(\frac{1.015075125 - 1}{0.005} \approx 3.015\)Thus:\[I_3 = 100 \times (3.015 - 3) = 1.5\]

Step 5: Calculate the Fourth Term (\(n=4\))

Substitute \(n = 4\) into the formula. \[I_4 = 100\left(\frac{1.005^4-1}{0.005} - 4\right)\]Calculate the expression:\(1.005^4 \approx 1.02015150125\)\(\frac{1.02015150125 - 1}{0.005} \approx 4.030\)Thus:\[I_4 = 100 \times (4.030 - 4) = 3\]

Step 6: Calculate the Fifth Term (\(n=5\))

Substitute \(n = 5\) into the formula. \[I_5 = 100\left(\frac{1.005^5-1}{0.005} - 5\right)\]Calculate the expression:\(1.005^5 \approx 1.02525332751875\)\(\frac{1.02525332751875 - 1}{0.005} \approx 5.0507\)Thus:\[I_5 = 100 \times (5.0507 - 5) = 5.07\]

Step 7: Calculate the Sixth Term (\(n=6\))

Substitute \(n = 6\) into the formula. \[I_6 = 100\left(\frac{1.005^6-1}{0.005} - 6\right)\]Calculate the expression:\(1.005^6 \approx 1.0303790931563438\)\(\frac{1.0303790931563438 - 1}{0.005} \approx 6.0758\)Thus:\[I_6 = 100 \times (6.0758 - 6) = 7.58\]

Step 8: Calculate the Interest for 5 Years (\(n=60\))

Substitute \(n = 60\) into the formula to find the interest accumulated after 5 years (60 months). \[I_{60} = 100\left(\frac{1.005^{60}-1}{0.005} - 60\right)\]Calculate the expression with a calculator as it involves a large power:\(1.005^{60} \approx 1.3488501\)\(\frac{1.3488501 - 1}{0.005} \approx 69.77\)Thus:\[I_{60} = 100 \times (69.77 - 60) = 977\]

Key Concepts

Sequence Formula

When dealing with financial calculations, sequence formulas are essential as they help us understand how values change over time. In the case of Helen's interest, the sequence formula is built from the compound interest principles. This specific formula calculates the interest accumulated over time when regular deposits are made. The formula given is \(I_{n}=100\left(\frac{1.005^{n}-1}{0.005}-n\right)\), where \(n\) represents the number of months.

The formula can seem complex at first, but its breakdown reveals simplicity. It is a form of geometric sequence where each term describes the accumulated interest after each month \(n\).

The core part, \(\frac{1.005^{n}-1}{0.005}\), captures the interest earned each month by leveraging the power of compounding. The subtraction of \(n\) represents the adjustment for the total deposits made at the rate of $100 per month. Understanding this breakdown helps in predicting future interest accurately.

Compounded Monthly

Compounding monthly is a popular method used in financial mathematics that enhances the value of an investment by reinvesting earned interest. In Helen’s case, the account pays an annual interest rate of 6%, which is divided into 12 monthly periods, hence a monthly compounding scenario.

Monthly compounding means that interest is calculated and added to the principal balance every month. Effectively, this means that each month, not only does your original deposit earn interest but so does the interest accumulated up until that point. This leads to earning 'interest on interest,' which can significantly grow your investment over time.

• Advantages: Faster growth, as interest is compounded regularly.
• Common in savings accounts, loans, and other financial products.

Understanding monthly compounding is pivotal as this affects the overall growth and final sum of the investment after a certain period, particularly in long-term savings or investments.

Interest Calculation

Interest calculation, especially in the context of compound interest, involves determining the amount of interest earned over a specified period. The interest formula used in Helen's problem considers the intricate compounding effect. Each month, part of the interest earned is added back to the principal, leading to more significant future interest earnings.

To compute interest, you should focus on:

• Identifying the principal amount – the initial amount deposited.
• Understanding the interest rate and how often it compounds (monthly in this case).
• Using the formula \(I_{n}=100\left(\frac{1.005^{n}-1}{0.005}-n\right)\) by substituting each month \(n\) to find the exact interest up to that month.

This systematic approach enables accurate determination of how much interest you accumulate over time and assists in evaluating the effectiveness of investments.

Financial Mathematics

Financial mathematics, or quantitative finance, applies mathematical concepts to financial problems. These concepts allow us to solve problems related to savings, investments, loans, and more comprehensively. In Helen's savings plan, financial mathematics is used to visualize growth through compound interest.

Core aspects include:

• Analyzing scenarios using formulas to predict outcomes like interest accumulation.
• Understanding time value of money – the concept that money today is worth more than the same amount in the future due to potential earning capacity.
• Applying these principles to real-world financial situations for better financial decision-making.

Financial mathematics equips students with essential skills needed to understand and manage personal finances efficiently. Understanding how these mathematical principles apply to financial products can significantly aid in better planning and investment strategies.