Problem 83
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. table can't copy
Step-by-Step Solution
Verified Answer
The accumulated interests are: $152 for 1 year, $310 for 2 years, $472 for 3 years, $638 for 4 years, and $808 for 5 years.
1Step 1: Understanding the Formula
The formula provided to calculate the interest accumulated is:\[ I = 5000\left[(1.0025)^{12n} - 1\right] \]where \( I \) is the accumulated interest, \( 5000 \) is the initial investment, and the term \((1.0025)^{12n}\) represents the compounding effect over \(n\) years, considering monthly compounding.
2Step 2: Setting Up the Table
Let's create a table to calculate the accumulated interest \(I\) for different values of \(n\), such as \(n = 1, 2, 3, 4,\) and \(5\) years. We will substitute each value of \(n\) into the formula to find the corresponding interest.
3Step 3: Calculating Interest for 1 Year
Substitute \( n = 1 \) into the formula:\[ I = 5000\left[(1.0025)^{12 \times 1} - 1\right] \]Calculate \( (1.0025)^{12} \) using a calculator to get approximately 1.0304. Then calculate the interest:\[ I = 5000\left[1.0304 - 1\right] = 5000 \times 0.0304 = 152 \]
4Step 4: Calculating Interest for 2 Years
Substitute \( n = 2 \) into the formula:\[ I = 5000\left[(1.0025)^{12 \times 2} - 1\right] \]Calculate \((1.0025)^{24}\) to get approximately 1.0620. Then calculate:\[ I = 5000\left[1.0620 - 1\right] = 5000 \times 0.0620 = 310 \]
5Step 5: Calculating Interest for 3 Years
Substitute \( n = 3 \) into the formula:\[ I = 5000\left[(1.0025)^{12 \times 3} - 1\right] \]Calculate \((1.0025)^{36}\) to get approximately 1.0944. Then calculate:\[ I = 5000\left[1.0944 - 1\right] = 5000 \times 0.0944 = 472 \]
6Step 6: Calculating Interest for 4 Years
Substitute \( n = 4 \) into the formula:\[ I = 5000\left[(1.0025)^{12 \times 4} - 1\right] \]Calculate \((1.0025)^{48}\) to get approximately 1.1276. Then calculate:\[ I = 5000\left[1.1276 - 1\right] = 5000 \times 0.1276 = 638 \]
7Step 7: Calculating Interest for 5 Years
Substitute \( n = 5 \) into the formula:\[ I = 5000\left[(1.0025)^{12 \times 5} - 1\right] \]Calculate \((1.0025)^{60}\) to get approximately 1.1616. Then calculate:\[ I = 5000\left[1.1616 - 1\right] = 5000 \times 0.1616 = 808 \]
8Step 8: Completing the Table
Now, complete the table with calculations for each year:- \( n = 1 \): Interest = \(152\)- \( n = 2 \): Interest = \(310\)- \( n = 3 \): Interest = \(472\)- \( n = 4 \): Interest = \(638\)- \( n = 5 \): Interest = \(808\)
Key Concepts
Monthly CompoundingExponential GrowthInterest CalculationInvestment Growth
Monthly Compounding
Investing your money in a financial product that performs monthly compounding is a smart strategy to maximize your returns over time. When we talk about monthly compounding, we mean that the interest is calculated and added to your initial investment every month. This means you earn interest on your accumulated interest, not just on your initial investment.
For example, in the given exercise, the money is compounded monthly at a 3% annual rate. This results in a monthly interest rate because the annual rate is divided by 12 months, making it easier to see the growth of your investment with each passing month.
This frequent addition of interest allows your investment to grow at an accelerated rate due to the continuous compounding effect, which is why understanding monthly compounding is crucial for anyone investing in products like certificates of deposit (CDs).
For example, in the given exercise, the money is compounded monthly at a 3% annual rate. This results in a monthly interest rate because the annual rate is divided by 12 months, making it easier to see the growth of your investment with each passing month.
This frequent addition of interest allows your investment to grow at an accelerated rate due to the continuous compounding effect, which is why understanding monthly compounding is crucial for anyone investing in products like certificates of deposit (CDs).
Exponential Growth
Exponential growth in the context of investments refers to the increase in value where the rate of growth becomes ever faster in proportion to the growing total number. This is often visualized in the formula as \( (1 + \frac{r}{n})^{nt} \), where the term is raised to the power representing the number of compounding periods.
In our example, \( (1.0025)^{12n} \) signifies the exponential growth of the investment over time. The base, 1.0025, represents the monthly compounding factor, and the exponent 12n indicates the total compounding periods over n years.
The true nature of exponential growth is slow and steady at first, but as the investing timelines extend, the growth begins to accelerate. This property makes it highly advantageous for long-term savings and investment plans, like CDs.
In our example, \( (1.0025)^{12n} \) signifies the exponential growth of the investment over time. The base, 1.0025, represents the monthly compounding factor, and the exponent 12n indicates the total compounding periods over n years.
The true nature of exponential growth is slow and steady at first, but as the investing timelines extend, the growth begins to accelerate. This property makes it highly advantageous for long-term savings and investment plans, like CDs.
Interest Calculation
Calculating interest using the compound interest method involves several steps, where interest is calculated recursively on the existing balance including previous interest add-ons.
In our example, to calculate the interest, the important step is to replace n with the relevant number of years in the formula \[ I = 5000\left[(1.0025)^{12n} - 1\right] \]. You calculate \( (1.0025)^{12n} \) to find how much your initial investment grows due to compounding, and then subtract 1 to focus on just the interest part.
After multiplication with the initial sum of $5000, you derive the exact amount of interest earned over the time period. This method highlights how even small interest rates can result in significant gains through compounding.
In our example, to calculate the interest, the important step is to replace n with the relevant number of years in the formula \[ I = 5000\left[(1.0025)^{12n} - 1\right] \]. You calculate \( (1.0025)^{12n} \) to find how much your initial investment grows due to compounding, and then subtract 1 to focus on just the interest part.
After multiplication with the initial sum of $5000, you derive the exact amount of interest earned over the time period. This method highlights how even small interest rates can result in significant gains through compounding.
Investment Growth
Investment growth is the increase in your initial investment capital over time, resulting from the accumulation of interest. In this context, understanding how to maximize growth through compounding is crucial for achieving financial goals effectively.
Looking at our CD example, we started with a $5000 initial deposit, experiencing cumulative growth over five years due to the power of monthly compounding.
To visualize this, after 5 years, the investment doesn't just amount to the principal plus flat interests; it grows significantly more due to compounding, resulting in cumulative growth. Each year, the interest experiences further growth itself, demonstrating how your investment grows exponentially.
Looking at our CD example, we started with a $5000 initial deposit, experiencing cumulative growth over five years due to the power of monthly compounding.
To visualize this, after 5 years, the investment doesn't just amount to the principal plus flat interests; it grows significantly more due to compounding, resulting in cumulative growth. Each year, the interest experiences further growth itself, demonstrating how your investment grows exponentially.
- Year 1: $152 interest
- Year 2: $310 interest
- Year 3: $472 interest
- Year 4: $638 interest
- Year 5: $808 interest
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