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 you 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.
For Investment 1, the amount deposited at the beginning of each year is $4,000. So the model polynomial, C(x), for Investment 1 is:
C(x) = 4000x^7
For Investment 2, the amounts deposited at the beginning of each year are $2,000, $1,500, and $500. Using the given expression x = 1 + r, we can represent the model polynomial, C(x), for Investment 2 as:
C(x) = (2000x^7) + (1500x^5) + (500x^3)
To determine which investment option will result in more interest earned at the end of the 7-year term with an annual interest rate of 3.7 percent, we substitute x = 1 + 0.037 (interest rate expressed as a decimal) into the model polynomials and compare the calculated values.
For Investment 1:
C(x) = 4000(1 + 0.037)^7
C(x) ≈ 4000(1.037)^7
C(x) ≈ 4000(1.2961)
C(x) ≈ $5,184.40
For Investment 2:
C(x) = (2000(1 + 0.037)^7) + (1500(1 + 0.037)^5) + (500(1 + 0.037)^3)
C(x) ≈ (2000(1.037)^7) + (1500(1.037)^5) + (500(1.037)^3)
C(x) ≈ (2000(1.2961)) + (1500(1.191)} + (500(1.111))
C(x) ≈ $2,592.20 + $1,785.15 + $555.50
C(x) ≈ $4,932.85
Therefore, Investment 1 will result in more interest earned at the end of the 7-year term compared to Investment 2.
Based on this result, it is generally better to invest more at one time rather than divide the investment into smaller deposits each year. Investment 1 yielded a higher final amount even though the total deposit amount for Investment 2 was greater than that of Investment 1 ($4,000 vs $4,000 + $2,000 + $1,500 + $500 = $8,000). This is because Investment 1 allows the entire deposit to earn interest for the full 7-year term, while Investment 2 allows smaller deposits to only earn interest for a portion of the 7-year term. Therefore, consolidating the investment into a larger initial deposit can result in more interest earned in the long run.
For Investment 1: Deposit $4,000 at the beginning of the first year, the model polynomial C(x) can be written as:
C(x) = 4000 * x^7
For 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, the model polynomial C(x) can be written as:
C(x) = 2000 * x^7 + 1500 * x^5 + 500 * x^3
To compare which investment option will result in more interest earned at the end of the 7-year term with an annual interest rate of 3.7 percent, we can substitute x = 1 + r into the model polynomials and evaluate them.
For Investment 1:
C(1 + 0.037) = 4000 * (1 + 0.037)^7
For Investment 2:
C(1 + 0.037) = 2000 * (1 + 0.037)^7 + 1500 * (1 + 0.037)^5 + 500 * (1 + 0.037)^3
After substituting and evaluating both polynomials, we can compare the final amounts to determine which investment option will result in more interest earned.
Regarding whether it is better to invest more at one time or divide the investment into smaller deposits each year, it depends on various factors. In general, investing more at one time may result in higher overall returns as the entire amount is working to earn interest. However, dividing the investment into smaller deposits allows for periodic contributions, which can help in building discipline and consistency in saving. It also reduces the risk of investing a large sum at an unfavorable time. The decision ultimately depends on individual preferences, financial goals, and risk tolerance.
To create a model polynomial, C(x), for the two investment options, we need to understand how the investments grow over time using the formula x = 1 + r, where r is the interest rate paid each year.
Investment 1: Deposit $4,000 at the beginning of the first year.
For this investment option, there is only one deposit made at the beginning of the first year, so the model polynomial can be written as:
C(x) = 4000 * x^7 (since the investment is for 7 years)
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.
For this investment option, there are three deposits made at different times. We'll break it down into three separate parts:
First deposit: $2,000 at the beginning of the first year:
C1(x) = 2000 * x^7
Second deposit: $1,500 at the beginning of the third year:
C2(x) = 1500 * x^5
Third deposit: $500 at the beginning of the fifth year:
C3(x) = 500 * x^3
To find the total amount for investment 2, we need to add the three separate parts:
C(x) = C1(x) + C2(x) + C3(x)
Now, we need to determine 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.
Let's substitute x = 1 + 0.037 (since the interest rate is 3.7%) into the model polynomials for both investment options:
For Investment 1: C(x) = 4000 * (1 + 0.037)^7
For Investment 2: C(x) = (2000 * (1 + 0.037)^7) + (1500 * (1 + 0.037)^5) + (500 * (1 + 0.037)^3)
Calculating these expressions will give us the final amounts for both investment options after 7 years.
Now, based on the answer you obtained for this problem, compare the final amounts from both investment options.
If investing more at one time resulted in a higher final amount, it suggests that making a larger initial deposit is beneficial. This is because the larger deposit has more time to earn interest over the 7-year period.
However, if dividing the investment into smaller deposits each year resulted in a higher final amount, it indicates that the compounding effect of making multiple deposits at different times can be advantageous.
Considering the calculations and results, you can determine whether it is better to invest more at one time or divide the investment into smaller deposits each year in different scenarios.