Problem 105 - Algebra and Trigonometry

Question

Interest on a CD A sum of \(\$ 5000\) is invested in a 5 -year certificate of deposit paying 3\(\%\) interest per year, compounded monthly. After \(n\) years the amount of interest \(I\) that has accumulated is given by $$I=5000\left[(1.0025)^{12 n}-1\right]$$ Complete the following table, which gives the amount of interest accumulated after the given number of years.

Step-by-Step Solution

Answer

The interest accumulated after 5 years is approximately $812.51.

Step 1: Understand the Formula

The formula given is \( I = 5000 \left[(1.0025)^{12n} - 1\right] \), where \( I \) represents the accumulated interest, and \( n \) is the number of years the investment is held.

Step 2: Calculate Interest for n = 1 Year

Substitute \( n = 1 \) into the formula: \[ I = 5000 \left[(1.0025)^{12 \times 1} - 1\right] \]Find \( (1.0025)^{12} \approx 1.030416 \). Then, \( I = 5000 \times (1.030416 - 1) = 5000 \times 0.030416 \). Calculate \( I \approx 152.08 \).

Step 3: Calculate Interest for n = 2 Years

Substitute \( n = 2 \) into the formula: \[ I = 5000 \left[(1.0025)^{12 \times 2} - 1\right] \]Find \( (1.0025)^{24} \approx 1.061836 \). Then, \( I = 5000 \times (1.061836 - 1) = 5000 \times 0.061836 \). Calculate \( I \approx 309.18 \).

Step 4: Calculate Interest for n = 3 Years

Substitute \( n = 3 \) into the formula: \[ I = 5000 \left[(1.0025)^{12 \times 3} - 1\right] \]Find \( (1.0025)^{36} \approx 1.094309 \). Then, \( I = 5000 \times (1.094309 - 1) = 5000 \times 0.094309 \). Calculate \( I \approx 471.55 \).

Step 5: Calculate Interest for n = 4 Years

Substitute \( n = 4 \) into the formula: \[ I = 5000 \left[(1.0025)^{12 \times 4} - 1\right] \]Find \( (1.0025)^{48} \approx 1.127860 \). Then, \( I = 5000 \times (1.127860 - 1) = 5000 \times 0.127860 \). Calculate \( I \approx 639.30 \).

Step 6: Calculate Interest for n = 5 Years

Substitute \( n = 5 \) into the formula: \[ I = 5000 \left[(1.0025)^{12 \times 5} - 1\right] \]Find \( (1.0025)^{60} \approx 1.162501 \). Then, \( I = 5000 \times (1.162501 - 1) = 5000 \times 0.162501 \). Calculate \( I \approx 812.505 \).

Step 7: Complete the Table

Using the calculations from Steps 2-6, the accumulated interest for each year is as follows: - After 1 year: approximately \( 152.08 \) dollars. - After 2 years: approximately \( 309.18 \) dollars. - After 3 years: approximately \( 471.55 \) dollars. - After 4 years: approximately \( 639.30 \) dollars. - After 5 years: approximately \( 812.505 \) dollars.

Key Concepts

Interest Calculation

Calculating interest, especially compound interest, involves understanding how interest is added to the principal amount over time. With compound interest, you not only earn interest on the initial principal but also on the accumulated interest from previous periods. In this particular exercise, the interest is compounded monthly, not annually.
This means interest is calculated twelve times a year, leading to more frequent compounding and slightly higher total interest earned over the same period than if it were compounded annually.
To calculate the interest accumulated over a given timeframe, the formula used is:

• \[ I = 5000 \left[(1.0025)^{12n} - 1\right] \]

Here, 5000 is the initial investment, 1.0025 is the monthly interest rate (3% annual rate divided by 12 months), and \( n \) represents the number of years. By inserting different values of \( n \), you can predict how much interest you will earn each year.

Investment Growth

Investment growth through compound interest represents a powerful way to expand your savings over time. In contrast to simple interest, where you earn returns only on your initial deposit, compound interest ensures your investment grows incrementally as interest is reinvested to generate even more interest in subsequent periods.
For example, in the exercise we considered:

• After 1 year, the interest added was approximately $152.08.
• By the end of the 5 years, it reached around $812.505.

By keeping your money in a compound interest yielding instrument, you allow the power of exponential growth to work its magic. The longer you let your funds sit and grow, the more pronounced the growth becomes, due to that compounding effect.

Exponential Growth

Exponential growth is an essential concept in understanding how compound interest works. With each compounding period, the interest is calculated not just on the original principal but also on the accumulated interest, leading to what is known as "growth on growth".
The compound interest formula: \((1.0025)^{12n}\) highlights this process. For every additional compounding period, the amount increases by the factor \(1.0025\).
Consider these examples:

• In the first year, the interest is a small portion, about 3% of the principal.
• However, by the fifth year, you've not only gained more interest each year but the total accumulated interest becomes a more significant portion of the initial investment.

The beauty of exponential growth in the context of investments is that time becomes your greatest ally. The earlier you invest, and the longer you stay invested, the more exponential growth can enhance your wealth.