Use x=1+r , where r is the interest rate paid each year. Write a model polynomial, C(x) , to represent the final amount of each of the following 7-year term investments.

Question

Investment 1: Deposit 4,000atthebeginningofthefirstyear.∗∗Investment2:∗∗Deposit 2,000 at the beginning of the first year, 1,500atthebeginningofthethirdyear,and 500 at the beginning of the fifth year. Using the models that you just created, which investment option will result in more interest earned at the end of the 7-year term if the annual interest rate is 3.7 percent? Based on the answer you got for this problem, when having the same interest rate and investment term, do you think it will be better to invest more at one time or divide the investment into smaller deposits each year? Explain your reasoning. Write it like a 10th grader

Answer 1

To solve the problem, we will create polynomial models representing the final amounts from both investments over a 7-year term with an interest rate of 3.7% per year.

Investment 1:

For Investment 1, you deposit $4,000 at the beginning of the first year. The formula for the future amount in years is given by \(C(x) = P \cdot x^n\), where \(P\) is the principal amount, \(x = 1 + r\), and \(n\) is the number of years the money is invested.

Here, \(r = 0.037\). So, \(x = 1 + 0.037 = 1.037\).

For Investment 1:

• Principal \(P = 4000\)
• Years \(n = 7\)

The future value will be: \[ C_1(x) = 4000 \cdot (1.037)^7 \]

Calculating \( (1.037)^7 \): \[ (1.037)^7 \approx 1.2819 \] Thus: \[ C_1(x) \approx 4000 \cdot 1.2819 \approx 5127.60 \]

Investment 2:

For Investment 2, deposits are made at different times:

• $2,000 at the beginning of Year 1 (invested for 7 years)
• $1,500 at the beginning of Year 3 (invested for 5 years)
• $500 at the beginning of Year 5 (invested for 3 years)

1. First deposit: \[ C_2(x) = 2000 \cdot (1.037)^7 \] \[ C_2(x) \approx 2000 \cdot 1.2819 \approx 5127.60 \]

2. Second deposit: \[ C_3(x) = 1500 \cdot (1.037)^5 \] Calculating \( (1.037)^5 \): \[ (1.037)^5 \approx 1.2004 \] Thus: \[ C_3(x) \approx 1500 \cdot 1.2004 \approx 1800.60 \]

3. Third deposit: \[ C_4(x) = 500 \cdot (1.037)^3 \] Calculating \( (1.037)^3 \): \[ (1.037)^3 \approx 1.1145 \] Thus: \[ C_4(x) \approx 500 \cdot 1.1145 \approx 557.25 \]

Total for Investment 2:

Now, we add up all amounts for Investment 2: \[ C_2(x) + C_3(x) + C_4(x) \approx 5127.60 + 1800.60 + 557.25 \approx 7485.45 \]

Comparison of Investments:

• Investment 1 total amount: $5,127.60
• Investment 2 total amount: $7,485.45

Conclusion: Investment 2 yields more money at the end of the 7 years.

Reasoning:

When considering whether to invest a lump sum or break it into smaller deposits, it’s generally better to invest more at one time if you can, because the money has a longer time to earn interest. In Investment 2, while smaller amounts were invested over time, the earlier you invest, the more interest you earn, especially if the interest rate is compounding. So, in a situation like this where you have the option, it's typically better to invest a larger amount upfront to maximize your returns!