Problem 72 - 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

(a) 0, 0.5, 1.5075, 2.020, 2.53, 3.0379; (b) 977.0023

Step 1: Understanding the Formula

The formula given to calculate the interest is \(I_{n}=100\left(\frac{1.005^{n}-1}{0.005}-n\right)\). It is used to compute the interest accumulated after depositing \$100 at the end of each month for \(n\) months, where the annual interest rate is 6\% compounded monthly. We derive each term by substituting specific values of \(n\).

Step 2: Calculating the First Term

Substitute \(n = 1\) into the formula to find the first term: \[ I_{1} = 100\left(\frac{1.005^{1}-1}{0.005}-1\right) \] Evaluate the expression:\[ I_{1} = 100\left(\frac{0.005}{0.005}-1\right) = 100(1-1) = 100 \times 0 = 0 \]

Step 3: Calculating the Second Term

Substitute \(n = 2\) into the formula to find the second term: \[ I_{2} = 100\left(\frac{1.005^{2}-1}{0.005}-2\right) \] Calculate the power and substitution:\[ I_{2} = 100\left(\frac{1.010025-1}{0.005}-2\right) = 100\left(2.005-2\right) = 100 \times 0.005 = 0.5 \]

Step 4: Calculating the Third Term

Substitute \(n = 3\) into the formula: \[ I_{3} = 100\left(\frac{1.005^{3}-1}{0.005}-3\right) \] Calculate the expression:\[ I_{3} = 100\left(\frac{1.015075125-1}{0.005}-3\right) = 100\left(3.015075125-3\right) = 100 \times 0.015075125 = 1.5075 \]

Step 5: Calculating the Fourth Term

Substitute \(n = 4\) into the formula: \[ I_{4} = 100\left(\frac{1.005^{4}-1}{0.005}-4\right) \] Calculate the result:\[ I_{4} = 100\left(\frac{1.020200376-1}{0.005}-4\right) = 100\left(4.020200376-4\right) = 100 \times 0.020200376 = 2.020 \]

Step 6: Calculating the Fifth Term

Substitute \(n = 5\) into the formula: \[ I_{5} = 100\left(\frac{1.005^{5}-1}{0.005}-5\right) \] Compute the term:\[ I_{5} = 100\left(\frac{1.025301878-1}{0.005}-5\right) = 100\left(5.025301878-5\right) = 100 \times 0.025301878 = 2.53 \]

Step 7: Calculating the Sixth Term

Substitute \(n = 6\) into the formula: \[ I_{6} = 100\left(\frac{1.005^{6}-1}{0.005}-6\right) \]Evaluate to get:\[ I_{6} = 100\left(\frac{1.030378884-1}{0.005}-6\right) = 100\left(6.030378884-6\right) = 100 \times 0.030378884 = 3.0379 \]

Step 8: Finding Interest After 5 Years

To find the interest accumulated after 5 years, calculate \(n = 5 \times 12 = 60\). Use the formula:\[ I_{60} = 100\left(\frac{1.005^{60}-1}{0.005}-60\right) \] Solve this expression:\[ I_{60} = 100\left(\frac{1.348850115-1}{0.005}-60\right) = 100\left(69.770023-60\right) = 100 \times 9.770023 = 977.0023 \]

Key Concepts

Interest Calculation

Interest calculation is a fundamental component of personal finance and investment. There are various ways to calculate interest, but in the context of Helen's problem, we are focusing on compound interest.
Compound interest is calculated on the initial principal and also on the accumulated interest from previous periods. It is usually more profitable than simple interest, where you only earn interest on the principal amount.
In Helen's case, interest is compounded monthly, meaning that each month, interest is calculated based on the total amount in the account at the end of the previous month. This includes both her monthly deposits and any interest that has been added to the initial balance.
Mathematically, the formula provided in the problem represents the accumulated interest over a specific time period:\[I_{n}=100\left(\frac{1.005^{n}-1}{0.005}-n\right)\]This formula considers a monthly deposit of $100 and a monthly interest rate converted from the annual rate of 6\%. Each term of the sequence gives the accumulated interest up to that particular month.

Monthly Deposits

Monthly deposits are regular contributions made to a financial account. They play a crucial role in the growth of savings and investments, especially when combined with compound interest.
In the scenario of Helen's savings, she deposits \(\$100\) at the end of each month. The timing of monthly deposits can affect the amount of interest accumulated, illustrating how regular contributions can lead to exponential growth over time.
With each deposit, the balance increases, and consequently, the interest for the following month is calculated on a larger amount. This demonstrates the power of consistent saving and investing. By systematically setting aside money every month, one can effectively leverage the principle of compound interest to accumulate wealth.

Sequence Evaluation

Sequence evaluation in mathematics is used to analyze the progression of numbers that follow a specific rule or pattern. The exercise with Helen's deposits translates into evaluating a sequence of interest amounts, identified as \(I_n\).
Each term in a sequence offers insight into the financial growth over successive months. In Helen's case, the sequence \(I_{n}=100\left(\frac{1.005^{n}-1}{0.005}-n\right)\) provides the accumulated interest for each month. To find the first six terms, you calculate the interest for different values of \(n\), where \(n\) represents the number of months.
For example, when \(n=1\), the first term of the sequence shows the accumulated interest after one month, which is \(0\) in this particular problem, indicating no interest earned yet on the first deposit. Continuing this at \(n=2, 3, 4, 5,\) and \(6\) months gives a snapshot of how interest accumulates initially. Such evaluations help in understanding the compounding effect over short intervals.

Time Value of Money

The time value of money (TVM) is a fundamental financial concept that explains why a specific amount of money today is worth more than the same amount in the future. This happens due to the potential earning capacity of the money when it is invested.
In Helen's situation, the sequence and the formula used serve as an example of TVM in action. By understanding how money grows over time with regular deposits and compounded interest, one can appreciate the benefits of saving early and consistently.
The compounded interest formula used in the exercise accounts for the future value of money, illustrating TVM by showing how future interest accumulates on today’s investments. When calculating interest after 5 years, it becomes clear that the earlier money is deposited and earns interest, the larger the accumulated amount will be.
This principle emphasizes making informed financial decisions early to maximize future financial health. By using the TVM principle, individuals like Helen can plan effectively for their financial goals, ensuring that money is working optimally over time.