Use x=1+r

Question

, 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.

Investment 1: Deposit $4,000 at the beginning of the first year.

Investment 2: Deposit $2,000 at the beginning of the first year, $1,500 at the beginning of the third year, 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.

Answer 1

To solve this problem, we will create the polynomial models for each investment based on the formula \( x = 1 + r \).

Given:

• Annual interest rate \( r = 0.037 \) (3.7%)
• \( x = 1 + r = 1 + 0.037 = 1.037 \)

Investment 1:

Deposit: $4,000 at the beginning of the first year for the entire 7-year term.

The final amount for Investment 1 can be calculated as: \[ C_1(x) = 4000 \cdot x^7 \] Substituting \( x \): \[ C_1(x) = 4000 \cdot (1.037)^7 \]

Investment 2:

Deposits:

1. $2,000 at the beginning of the first year.
2. $1,500 at the beginning of the third year.
3. $500 at the beginning of the fifth year.

The final amount for Investment 2 can be calculated by summing each deposit’s future value:

1. The $2,000 for 7 years: \[ 2000 \cdot x^7 \]

2. The $1,500 for 5 years (since it is deposited at the beginning of the 3rd year): \[ 1500 \cdot x^5 \]

3. The $500 for 3 years (since it is deposited at the beginning of the 5th year): \[ 500 \cdot x^3 \]

Thus, the final amount for Investment 2 is: \[ C_2(x) = 2000 \cdot x^7 + 1500 \cdot x^5 + 500 \cdot x^3 \]

Final Amounts Calculation:

Now we need to plug in the value of \( x = 1.037 \) to calculate the final amounts.

For Investment 1: \[ C_1(1.037) = 4000 \cdot (1.037)^7 \approx 4000 \cdot 1.28368 \approx 5134.72 \]

For Investment 2: \[ C_2(1.037) = 2000 \cdot (1.037)^7 + 1500 \cdot (1.037)^5 + 500 \cdot (1.037)^3 \] Calculating each component:

• \( 2000 \cdot (1.037)^7 \approx 2000 \cdot 1.28368 \approx 2567.36 \)
• \( 1500 \cdot (1.037)^5 \approx 1500 \cdot 1.19258 \approx 1788.87 \)
• \( 500 \cdot (1.037)^3 \approx 500 \cdot 1.11364 \approx 556.82 \)

Combining these: \[ C_2(1.037) = 2567.36 + 1788.87 + 556.82 \approx 4913.05 \]

Total Interest Earned:

• Investment 1 Final Amount: $5,134.72
• Investment 2 Final Amount: $4,913.05

To find out how much interest was earned:

• Investment 1:
  • Interest: \( 5134.72 - 4000 = 1134.72 \)
• Investment 2:
  • Interest: \( 4913.05 - (2000 + 1500 + 500) = 4913.05 - 4000 = 913.05 \)

Conclusion:

Investment 1 will result in more interest earned.

Final Question:

When having the same interest rate and investment term, it is generally better to invest a larger amount upfront rather than dividing the investment into smaller deposits. This is because the larger initial deposit benefits from compound interest for the full investment term, while later deposits earn interest for a shorter period. In this specific case, Investment 1 (larger upfront deposit) resulted in a greater total return compared to Investment 2 (smaller incremental deposits).

Answer 2

write in a paragraph

Answer 3

For a 7-year investment term at an annual interest rate of 3.7%, two investment strategies were compared: one involved a single deposit of $4,000 at the beginning of the first year, and the other consisted of three deposits totaling $4,000—$2,000 at the start, $1,500 at the beginning of the third year, and $500 at the beginning of the fifth year. Using the formula \( x = 1 + r \), where \( x = 1.037 \), the final amount for the first investment was approximately $5,134.72, yielding an interest of about $1,134.72, while the second investment accumulated to around $4,913.05, resulting in an interest gain of about $913.05. Consequently, the first investment option generated more interest due to the benefits of compounding over the full investment term. This analysis implies that, when faced with the same interest rate and investment duration, it is generally more advantageous to invest a larger sum upfront versus making smaller, staggered deposits, as the larger initial investment takes full advantage of compound interest throughout the entire period.