Question
use , where r is the interest rate paid each year. Write a model polynomial,
, 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.
x = 1 + r 𝑥 = 1 + 𝑟
C(x) 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.
x = 1 + r 𝑥 = 1 + 𝑟
C(x) 𝐶(𝑥)
Answers
GPT-4o mini
To model the final amount of each investment as a polynomial, we'll use the formula for compound interest, which can be expressed in terms of the interest rate \( r \) and the number of years until the amount is withdrawn or evaluated.
### Investment 1
**Investment 1:** Deposit $4,000 at the beginning of the first year for 7 years.
The future value \( C_1(x) \) of the first investment can be represented as follows:
\[
C_1(x) = 4000 \cdot x^7
\]
where \( x = 1 + r \).
### Investment 2
**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 future value \( C_2(x) \) of the second investment can be represented as:
- The first deposit of $2,000 grows for 7 years: \( 2000 \cdot x^7 \)
- The second deposit of $1,500 grows for 5 years: \( 1500 \cdot x^5 \)
- The third deposit of $500 grows for 3 years: \( 500 \cdot x^3 \)
Combining these, we have:
\[
C_2(x) = 2000 \cdot x^7 + 1500 \cdot x^5 + 500 \cdot x^3
\]
### Substitute \( r = 0.037 \)
Since the interest rate \( r \) is 3.7 percent, we can compute \( x \):
\[
x = 1 + r = 1 + 0.037 = 1.037
\]
### Calculate \( C_1(x) \)
Now, substitute \( x \) into the formula for Investment 1:
\[
C_1(1.037) = 4000 \cdot (1.037)^7
\]
Calculating \( (1.037)^7 \):
\[
(1.037)^7 \approx 1.292537
\]
Now:
\[
C_1(1.037) \approx 4000 \cdot 1.292537 \approx 5170.15
\]
### Calculate \( C_2(x) \)
Now, substitute \( x \) into the formula for Investment 2:
\[
C_2(1.037) = 2000 \cdot (1.037)^7 + 1500 \cdot (1.037)^5 + 500 \cdot (1.037)^3
\]
We already calculated \( (1.037)^7 \). Now we need to calculate \( (1.037)^5 \) and \( (1.037)^3 \):
\[
(1.037)^5 \approx 1.199679
\]
\[
(1.037)^3 \approx 1.113364
\]
Putting it all together:
\[
C_2(1.037) \approx 2000 \cdot 1.292537 + 1500 \cdot 1.199679 + 500 \cdot 1.113364
\]
Calculating:
\[
C_2(1.037) \approx 2585.07 + 1799.52 + 556.682 \approx 4941.27
\]
### Interest Earned
Now we can calculate the interest earned for both investments by subtracting the initial deposit from the future value.
- For Investment 1:
\[
\text{Interest}_1 = C_1(1.037) - 4000 \approx 5170.15 - 4000 = 1170.15
\]
- For Investment 2:
\[
\text{Interest}_2 = C_2(1.037) - (2000 + 1500 + 500) = 4941.27 - 4000 \approx 941.27
\]
### Conclusion
Based on the calculations:
- **Interest from Investment 1:** \( 1170.15 \)
- **Interest from Investment 2:** \( 941.27 \)
Thus, Investment 1 earns more interest.
### General Investment Strategy
When choosing between making a one-time larger investment versus smaller periodic investments:
- In general, investing a larger amount at one time can yield higher returns if the investment period allows for compounding to occur fully over that duration.
- However, the choice may also depend on personal cash flow considerations, risk tolerance, and market conditions.
In this scenario, because both investments were for the same duration and faced the same interest rate, the one-time larger investment clearly resulted in higher total interest, underscoring the potential advantage of lump-sum investing in a stable interest rate environment.
### Investment 1
**Investment 1:** Deposit $4,000 at the beginning of the first year for 7 years.
The future value \( C_1(x) \) of the first investment can be represented as follows:
\[
C_1(x) = 4000 \cdot x^7
\]
where \( x = 1 + r \).
### Investment 2
**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 future value \( C_2(x) \) of the second investment can be represented as:
- The first deposit of $2,000 grows for 7 years: \( 2000 \cdot x^7 \)
- The second deposit of $1,500 grows for 5 years: \( 1500 \cdot x^5 \)
- The third deposit of $500 grows for 3 years: \( 500 \cdot x^3 \)
Combining these, we have:
\[
C_2(x) = 2000 \cdot x^7 + 1500 \cdot x^5 + 500 \cdot x^3
\]
### Substitute \( r = 0.037 \)
Since the interest rate \( r \) is 3.7 percent, we can compute \( x \):
\[
x = 1 + r = 1 + 0.037 = 1.037
\]
### Calculate \( C_1(x) \)
Now, substitute \( x \) into the formula for Investment 1:
\[
C_1(1.037) = 4000 \cdot (1.037)^7
\]
Calculating \( (1.037)^7 \):
\[
(1.037)^7 \approx 1.292537
\]
Now:
\[
C_1(1.037) \approx 4000 \cdot 1.292537 \approx 5170.15
\]
### Calculate \( C_2(x) \)
Now, substitute \( x \) into the formula for Investment 2:
\[
C_2(1.037) = 2000 \cdot (1.037)^7 + 1500 \cdot (1.037)^5 + 500 \cdot (1.037)^3
\]
We already calculated \( (1.037)^7 \). Now we need to calculate \( (1.037)^5 \) and \( (1.037)^3 \):
\[
(1.037)^5 \approx 1.199679
\]
\[
(1.037)^3 \approx 1.113364
\]
Putting it all together:
\[
C_2(1.037) \approx 2000 \cdot 1.292537 + 1500 \cdot 1.199679 + 500 \cdot 1.113364
\]
Calculating:
\[
C_2(1.037) \approx 2585.07 + 1799.52 + 556.682 \approx 4941.27
\]
### Interest Earned
Now we can calculate the interest earned for both investments by subtracting the initial deposit from the future value.
- For Investment 1:
\[
\text{Interest}_1 = C_1(1.037) - 4000 \approx 5170.15 - 4000 = 1170.15
\]
- For Investment 2:
\[
\text{Interest}_2 = C_2(1.037) - (2000 + 1500 + 500) = 4941.27 - 4000 \approx 941.27
\]
### Conclusion
Based on the calculations:
- **Interest from Investment 1:** \( 1170.15 \)
- **Interest from Investment 2:** \( 941.27 \)
Thus, Investment 1 earns more interest.
### General Investment Strategy
When choosing between making a one-time larger investment versus smaller periodic investments:
- In general, investing a larger amount at one time can yield higher returns if the investment period allows for compounding to occur fully over that duration.
- However, the choice may also depend on personal cash flow considerations, risk tolerance, and market conditions.
In this scenario, because both investments were for the same duration and faced the same interest rate, the one-time larger investment clearly resulted in higher total interest, underscoring the potential advantage of lump-sum investing in a stable interest rate environment.