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.
To find the model polynomial for each investment option, we can use the formula C(x) = P(1 + r)^n, where C(x) is the final amount, P is the initial deposit, r is the interest rate, and n is the number of years.
Investment 1:
P = $4,000
n = 7 years
r = 3.7% = 0.037 (decimal form of interest rate)
C(x) = 4000(1 + 0.037)^7
Investment 2:
For this investment, we need to find the final amount for each deposit separately and then add them together.
First deposit:
P1 = $2,000
n1 = 7 years
C1(x) = 2000(1 + 0.037)^7
Second deposit:
P2 = $1,500
n2 = 7 - 3 = 4 years (beginning of the third year to the end of the seventh year)
C2(x) = 1500(1 + 0.037)^4
Third deposit:
P3 = $500
n3 = 7 - 5 = 2 years (beginning of the fifth year to the end of the seventh year)
C3(x) = 500(1 + 0.037)^2
C(x) = C1(x) + C2(x) + C3(x)
Now let's calculate the final amounts for each investment option to compare them.
Investment 1:
C(x) = 4000(1 + 0.037)^7 ≈ $4,937.71
Investment 2:
C1(x) = 2000(1 + 0.037)^7 ≈ $2,468.86
C2(x) = 1500(1 + 0.037)^4 ≈ $1,715.69
C3(x) = 500(1 + 0.037)^2 ≈ $520.40
C(x) = C1(x) + C2(x) + C3(x) ≈ $4,705.95
Therefore, Investment 1 will result in more interest earned at the end of the 7-year term if the annual interest rate is 3.7 percent.
When having the same interest rate and investment term, it is generally better to invest more at one time. This is because compounding interest works better with larger initial deposits. The more money you have invested initially, the more interest you will earn over time. Additionally, investing more at one time also simplifies the process and requires less effort to manage multiple deposits.
For Investment 1: Deposit $4,000 at the beginning of the first year, the model polynomial C(x) would be:
C(x) = 4000 * (1+r)^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) would be:
C(x) = 2000 * (1+r)^7 + 1500 * (1+r)^5 + 500 * (1+r)^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 can evaluate C(x) for both investments when x=1.037 (1+r):
For Investment 1:
C(1.037) = 4000 * (1.037)^7 ≈ $4,681.48
For Investment 2:
C(1.037) = 2000 * (1.037)^7 + 1500 * (1.037)^5 + 500 * (1.037)^3 ≈ $4,900.95
Therefore, Investment 2 will result in more interest earned at the end of the 7-year term if the annual interest rate is 3.7 percent.
When comparing the two investment options with the same interest rate and investment term, it is generally better to invest more at one time rather than dividing the investment into smaller deposits each year. In this case, Investment 2, with staggered deposits, resulted in slightly higher interest earnings. However, this may not always be the case as it depends on various factors such as interest rates, compounding periods, and the timing of deposits. Investing more at one time reduces the impact of compounding on smaller deposits and typically leads to higher overall returns.