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.
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.
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