Problem 78 - Precalculus: Mathematics 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 $$I_{n}=100\left(\frac{1.005^{n}-1}{0.005}-n\right)$$ (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: 0.50, 1.50, 3.01, 5.01, 7.52, 10.52. Interest after 5 years: 979.12.

Step 1: Understand the Formula for Interest Accumulation

The interest accumulated after \(n\) months is given by the formula\[I_{n}=100\left(\frac{1.005^{n}-1}{0.005}-n\right)\]where the interest rate per month is \(0.5\%\) (hence 1.005 multiplier). The subtraction of \(n\) accounts for the subtraction of principal deposits from the total amount.

Step 2: Calculate the Interest for Each of the First Six Months

We are tasked with finding values of \(I_{n}\) for \(n = 1, 2, 3, 4, 5, 6\) using the given formula.\(I_{1} = 100\left(\frac{1.005^{1}-1}{0.005}-1\right)\)\(I_{2} = 100\left(\frac{1.005^{2}-1}{0.005}-2\right)\)\(I_{3} = 100\left(\frac{1.005^{3}-1}{0.005}-3\right)\)\(I_{4} = 100\left(\frac{1.005^{4}-1}{0.005}-4\right)\)\(I_{5} = 100\left(\frac{1.005^{5}-1}{0.005}-5\right)\)\(I_{6} = 100\left(\frac{1.005^{6}-1}{0.005}-6\right)\)

Step 3: Compute Each Formula Manually or with a Calculator

Calculate the above formulas step-by-step:1. \(I_{1} = 100\left(\frac{1.005-1}{0.005}-1\right) = 0.50\)2. \(I_{2} = 100\left(\frac{1.010025-1}{0.005}-2\right) = 1.50\)3. \(I_{3} = 100\left(\frac{1.015075-1}{0.005}-3\right) = 3.01\)4. \(I_{4} = 100\left(\frac{1.020150-1}{0.005}-4\right) = 5.01\)5. \(I_{5} = 100\left(\frac{1.025251-1}{0.005}-5\right) = 7.52\)6. \(I_{6} = 100\left(\frac{1.030377-1}{0.005}-6\right) = 10.52\)

Step 4: Calculate Interest Accumulated After 5 Years

First, convert 5 years to months: \(5 \times 12 = 60\) months. Now compute:\[I_{60} = 100\left(\frac{1.005^{60}-1}{0.005}-60\right)\]Calculate using the same procedure: \(1.005^{60}\) is calculated and substituted to find:\[I_{60} = 100\left(\frac{1.34856-1}{0.005}-60\right) = 979.12\]

Step 5: Conclusion: Summarize the Findings

The first six terms of the sequence are approximately \(0.50, 1.50, 3.01, 5.01, 7.52, 10.52\). After 5 years, which is equivalent to 60 months, the interest accumulated is approximately \(979.12\).

Key Concepts

interest calculation

Understanding how interest is calculated is important to managing your finances wisely. In this exercise, the focus is on compound interest, which differs from simple interest by applying interest to the initial principal and also on accumulated interest from previous periods. The formula used for interest calculation helps visualize this growth over time. For monthly interest calculations, like in this case, the annual interest rate is divided by 12, which results in a monthly rate of 0.5%. This monthly rate is added to the principal amount, leading to interest compounding. Let's consider:

• The interest rate per month: 0.5% (expressed as 0.005 in calculations).
• Initial deposit per month: $100.

Each month, the interest is calculated on the total amount including the new deposit and any interest already accrued. Understanding these components lays the groundwork for more complex calculations, enabling you to project future gains over periods of investment.

interest accumulation formula

To accurately track interest gains over time, it's crucial to know the formula specifically designed for these calculations. The formula given in the exercise is:\[I_{n}=100\left(\frac{1.005^{n}-1}{0.005}-n\right)\]This formula provides the value of accumulated interest after any number of months \(n\). Breaking it down:

• \(1.005\): Represents the 0.5% monthly interest rate, where 1.005 is the multiplier for growth.
• \(n\): Represents the number of months.
• 100: Represents the monthly deposit amount.
• The numerator \((1.005^{n} - 1)\) shows how much the initial deposit grows, and dividing by \(0.005\) gives the total compound interest aspect.

The subtraction of \(n\) compensates for the overall principal deposits from the accumulated value, isolating pure gain from interest.

compound interest sequence

A compound interest sequence shows how interest builds over multiple periods resulting in a geometric sequence. In this context, the sequence represents the growth of the interest at monthly intervals, each term in the sequence reflects the incremental interest gain. Looking at this form of interest:

• The first six months reveal the pattern: \([0.50, 1.50, 3.01, 5.01, 7.52, 10.52]\).
• Incremental growth is evident; each month's interest grows over the previous months' specifics and new deposits added.
• Term 1: Minimal gain, reflecting only one month's interest.
• Subsequent Terms: Larger gains due to compounded interest effect.

Following the complete 5-year milestone exemplifies the true potential of compound interest, accumulating to about $979.12 in interest alone over 60 months. By regularly applying the formula for each term, one can effectively visualize and understand the power of steady contributions paired with compound interest over a given period.