Problem 70 - Algebra and Trigonometry

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

The first six terms are 0, 0.50, 1.51, 2.52, 3.53, 4.54. After 5 years, the accumulated interest is $288.50.

Step 1: Understand the Formula

We are given the sequence for accumulated interest: \[ I_{n} = 100\left(\frac{1.005^{n} - 1}{0.005} - n\right) \]This represents the accumulation of monthly deposits of \$100 compounded monthly at an annual interest rate of 6%, or 0.5% monthly.

Step 2: Calculate the First Term

For the first term when \( n = 1 \):\[ I_1 = 100\left(\frac{1.005^{1} - 1}{0.005} - 1\right) \]Simplifying, we find:\[ I_1 = 100\left(0.005 / 0.005 - 1\right) = 0 \]

Step 3: Calculate the Second Term

For \( n = 2 \):\[ I_2 = 100\left(\frac{1.005^{2} - 1}{0.005} - 2\right) \]Calculating further:\[ I_2 = 100\left(0.010025 / 0.005 - 2\right) = 0.50 \]

Step 4: Calculate the Third Term

For \( n = 3 \):\[ I_3 = 100\left(\frac{1.005^{3} - 1}{0.005} - 3\right) \]Calculate step-by-step:\[ I_3 = 100\left(0.015075125 / 0.005 - 3\right) = 1.51 \]

Step 5: Calculate the Fourth Term

For \( n = 4 \):\[ I_4 = 100\left(\frac{1.005^{4} - 1}{0.005} - 4\right) \]With computation:\[ I_4 = 100\left(0.020150626 / 0.005 - 4\right) = 2.52 \]

Step 6: Calculate the Fifth Term

For \( n = 5 \):\[ I_5 = 100\left(\frac{1.005^{5} - 1}{0.005} - 5\right) \]Evaluate as follows:\[ I_5 = 100\left(0.025251377 / 0.005 - 5\right) = 3.53 \]

Step 7: Calculate the Sixth Term

For \( n = 6 \):\[ I_6 = 100\left(\frac{1.005^{6} - 1}{0.005} - 6\right) \]Proceed to compute:\[ I_6 = 100\left(0.030377134 / 0.005 - 6\right) = 4.54 \]

Step 8: Calculate Interest After 5 Years

Helen deposits for 5 years or 60 months, so we use \( n = 60 \):\[ I_{60} = 100\left(\frac{1.005^{60} - 1}{0.005} - 60\right) \]Calculate the amounts:\[ I_{60} = 100\left(0.348850 / 0.005 - 60\right) = 288.50 \]

Key Concepts

Monthly Compounding

Monthly compounding is a method of calculating interest where the interest is added to the principal every month. This means that each month, the interest earned becomes part of the principal, which then earns additional interest in the next month.
For Helen, whose account earns 6% interest annually, the monthly interest rate would be 0.5%. This is because the annual rate is divided by 12 months. Each time Helen deposits her $100, it starts earning interest immediately, compounding the interest monthly.

• The frequency of compounding could significantly affect the amount accumulated over time.
• Monthly compounding offers a higher yield than yearly compounding because interest is paid more frequently.

Understanding this type of interest calculation helps us see why small monthly deposits can grow substantially over the years.

Interest Accumulation

Interest accumulation refers to the total amount of interest that has been added to the initial sum of money over time. In Helen's case, interest accumulation is calculated monthly, as each deposit she makes earns interest.
The formula given, \[ I_{n} = 100\left(\frac{1.005^{n} - 1}{0.005} - n\right) \]helps determine how much interest is accumulated after each month, subtracting the total deposited amount (which does not earn interest right away) from the compounded balance.

• The exponential growth factor, \(1.005^n\), reveals how the interest grows over time.
• "n" represents the number of months the money has been in the account.

This sequence shows not just the growth, but the power of compound interest over a period, demonstrating why saving even small amounts monthly can lead to considerable growth in savings.

Financial Mathematics

Financial mathematics involves using mathematical formulas and principles to assess investment and savings options. A basic understanding of financial mathematics can be extremely beneficial when it comes to managing and growing personal finances.
In Helen's scenario, she leverages financial mathematics by employing a sequence to predict her interest accumulation over time. The formula provided models how each deposit grows with each compounding period.

• Using sequences and growth formulas can foresee future amounts based on current trends.
• It helps in understanding the time value of money – the idea that money available now is worth more than the same sum in the future due to its potential earning capacity.

This knowledge allows individuals to make informed decisions about savings and investments, assuring better financial well-being over time.