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.

Question

GPT-4o mini · Answer

To solve this problem, we first need to define the polynomial models for each investment using the formula $ x = 1 + r $. Since the interest rate is given as 3.7%, we convert this to a decimal for our calculations:

\[ r = 0.037 \implies x = 1 + r = 1 + 0.037 = 1.037 \]

Investment 1:

For Investment 1, we deposit $4,000 at the beginning of the first year. The future value is based on compounding interest over 7 years. The polynomial model can be set up as follows:

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

Investment 2:

For Investment 2, we make multiple deposits at different times:

Deposit $2,000 at the beginning of the first year.
Deposit $1,500 at the beginning of the third year (which earns interest for 5 years).
Deposit $500 at the beginning of the fifth year (which earns interest for 3 years).

The polynomial model for Investment 2 can be expressed as:

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

Evaluating the Models:

Now we will substitute $ x = 1.037 $ into both models to find the final amounts.

For Investment 1:

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

Calculating $ (1.037)^7 $:

\[ (1.037)^7 \approx 1.28403 \quad \text{(using a calculator)} \]

So,

\[ C_1(1.037) \approx 4000 \cdot 1.28403 \approx 5136.12 \]

For Investment 2:

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

Calculating each term:

First term: $ 2000 \cdot (1.037)^7 \approx 2000 \cdot 1.28403 \approx 2568.06 $
Second term: $ 1500 \cdot (1.037)^5 $

Calculating $ (1.037)^5 $: \[ (1.037)^5 \approx 1.195618 \quad \text{(using a calculator)} \] So, \[ 1500 \cdot (1.037)^5 \approx 1500 \cdot 1.195618 \approx 1793.43 \]

Third term: $ 500 \cdot (1.037)^3 $

Calculating $ (1.037)^3 $: \[ (1.037)^3 \approx 1.114882 \quad \text{(using a calculator)} \] So, \[ 500 \cdot (1.037)^3 \approx 500 \cdot 1.114882 \approx 557.44 \]

Adding these together:

\[ C_2(1.037) \approx 2568.06 + 1793.43 + 557.44 \approx 4918.93 \]

Summary:

Final amount for Investment 1: $5,136.12
Final amount for Investment 2: $4,918.93

Interest Earned:

For Investment 1, interest earned = $ 5136.12 - 4000 = 1136.12 $
For Investment 2, interest earned = $ 4918.93 - 4000 = 918.93 $

Conclusion:

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

Reasoning:

In general, investing a larger amount at one time can lead to more interest earned than dividing the investment into smaller deposits. This is because money invested earlier has more time to grow due to compounding interest. In this case, even with smaller subsequent investments, the initial larger investment in Investment 1 allowed for more compounding growth over the entire term compared to the staggered deposits in Investment 2.

Answer

make the reasoning simpler

GPT-4o mini · Answer

Certainly! Here’s a simpler explanation:

Investment 1: You invest $4,000 all at once at the beginning of the 7 years. It earns interest the entire time.
Investment 2: You invest smaller amounts at different times: $2,000 at the start, $1,500 after 2 years, and $500 after 4 years. The smaller amounts have less time to earn interest.

Result:

Investment 1 grows to approximately $5,136.12, earning about $1,136.12 in interest.
Investment 2 grows to approximately $4,918.93, earning about $918.93 in interest.

Conclusion:

Investment 1 earns more interest because the full amount is invested longer.

Reasoning:

Investing a larger sum upfront is generally better because it has more time to grow due to compound interest. Smaller, staggered investments don't benefit as much from this. Therefore, if you can, it's usually better to invest a bigger amount all at once rather than spreading smaller deposits over time.