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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. 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.

1 answer

Let's start by defining the polynomial models for each investment option using the formula \( x = 1 + r \).

Given that the interest rate \( r = 0.037 \) (for 3.7%), we find \( x \):

\[ x = 1 + 0.037 = 1.037 \]

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

This $4,000 will compound for 7 years. The final amount, \( C_1(x) \) for Investment 1, can be represented as:

\[ C_1(x) = 4000 \cdot x^7 \]

Substituting \( x = 1.037 \):

\[ C_1(1.037) = 4000 \cdot (1.037)^7 \]

Calculating \( (1.037)^7 \):

\[ (1.037)^7 \approx 1.281574 \]

Now calculating \( C_1(1.037) \):

\[ C_1(1.037) \approx 4000 \cdot 1.281574 \approx 5126.30 \]

Investment 2: Deposits of $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.

For Investment 2, we will track each deposit separately based on how many years they will compound:

  1. The first deposit of $2,000 for 7 years:

\[ C_{2a}(x) = 2000 \cdot x^7 \]

  1. The second deposit of $1,500 for 5 years (deposited at the beginning of Year 3):

\[ C_{2b}(x) = 1500 \cdot x^5 \]

  1. The third deposit of $500 for 3 years (deposited at the beginning of Year 5):

\[ C_{2c}(x) = 500 \cdot x^3 \]

The total amount at the end of 7 years for Investment 2, \( C_2(x) \), is given by:

\[ C_2(x) = 2000 \cdot x^7 + 1500 \cdot x^5 + 500 \cdot x^3 \]

Substituting \( x = 1.037 \):

  1. Calculate \( (1.037)^5 \):

\[ (1.037)^5 \approx 1.20085 \]

  1. Calculate \( (1.037)^3 \):

\[ (1.037)^3 \approx 1.11576 \]

Now we can find \( C_2(1.037) \):

\[ C_2(1.037) = 2000 \cdot (1.037)^7 + 1500 \cdot (1.037)^5 + 500 \cdot (1.037)^3 \]

Substituting the values:

\[ C_2(1.037) \approx 2000 \cdot 1.281574 + 1500 \cdot 1.20085 + 500 \cdot 1.11576 \]

Calculating each term:

\[ 2000 \cdot 1.281574 \approx 2563.15 \] \[ 1500 \cdot 1.20085 \approx 1800.13 \] \[ 500 \cdot 1.11576 \approx 557.88 \]

Now summing the amounts:

\[ C_2(1.037) \approx 2563.15 + 1800.13 + 557.88 \approx 4910.16 \]

Summary of Investments:

  • Investment 1 Final Amount: \( C_1(1.037) \approx 5126.30 \)
  • Investment 2 Final Amount: \( C_2(1.037) \approx 4910.16 \)

Conclusion on Interest Earned

  • Interest Earned from Investment 1: \( 5126.30 - 4000 = 1126.30 \)
  • Interest Earned from Investment 2: \( 4910.16 - 4000 = 910.16 \)

Thus, Investment 1 results in more interest earned at the end of the 7-year term.

Whether to Invest More at Once or in Smaller Deposits

Investing a larger amount at once (like in Investment 1) resulted in higher accumulation of interest compared to breaking it down into smaller deposits over time (like in Investment 2). This is due to the effect of compounding interest on a larger principal amount for a longer period. The earlier the money is invested, the more it can grow due to compounding. Therefore, it is generally better to invest more at one time when access to funds allows for it and especially when the interest rate is favorable.