When an initial amount of è dollars is invested at r% annual interest compounded n times per year, the value of the account (4) after 1 years s given by the equation t r A=P(1+²)*** n nt Write an equation that represents the value in an account that starts out with an initial investment of $5000 and pays 10% interest compound monthly. Then use that equation to fill the table and use the table to graph the equation. B M
When an initial amount of è dollars is invested at r% annual interest compounded n times per year, the value of the account (4) after 1 years s given by the equation t r A=P(1+²)*** n nt Write an equation that represents the value in an account that starts out with an initial investment of $5000 and pays 10% interest compound monthly. Then use that equation to fill the table and use the table to graph the equation. B M
Essentials Of Investments
11th Edition
ISBN:9781260013924
Author:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Publisher:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Chapter1: Investments: Background And Issues
Section: Chapter Questions
Problem 1PS
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![### Understanding Compound Interest
When an initial amount of \( P \) dollars is invested at \( r \)% annual interest compounded \( n \) times per year, the value of the account (\( A \)) after \( t \) years is given by the equation:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
### Example Problem
Write an equation that represents the value in an account that starts out with an initial investment of $5000 and pays 10% interest compounded monthly. Then use that equation to fill the table and use the table to graph the equation.
### Solution
Given:
- Initial Investment (\( P \)) = $5000
- Annual Interest Rate (\( r \)) = 10% = 0.10
- Number of times interest is compounded per year (\( n \)) = 12 (monthly)
Substituting these values into the compound interest formula, we get:
\[ A = 5000 \left(1 + \frac{0.10}{12}\right)^{12t} \]
\[ A = 5000 \left(1 + \frac{0.10}{12}\right)^{12t} \]
\[ A = 5000 \left(1 + 0.00833\right)^{12t} \]
\[ A = 5000 (1.00833)^{12t} \]
### Fill in the Table
| Years (\( t \)) | Value (\( A \)) |
|-----------------|--------------------|
| 0 | $5000 |
| 5 | $ |
| 10 | $ |
| 15 | $ |
| 20 | $ |
To calculate the values, we plug in the corresponding \( t \) values into the equation:
For \( t = 5 \):
\[ A = 5000 (1.00833)^{12 \times 5} \]
For \( t = 10 \):
\[ A = 5000 (1.00833)^{12 \times 10} \]
For \( t = 15 \):
\[ A = 5000 (1.00833)^{12 \times 15} \]
For \( t = 20 \):
\[ A = 5000 (1.00833)^{12 \times](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf54cc92-fe70-42ec-a5a5-6c03b8447a69%2F95a06ca2-e50f-449d-8f4a-c6896b50efb9%2Fb9e19m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Compound Interest
When an initial amount of \( P \) dollars is invested at \( r \)% annual interest compounded \( n \) times per year, the value of the account (\( A \)) after \( t \) years is given by the equation:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
### Example Problem
Write an equation that represents the value in an account that starts out with an initial investment of $5000 and pays 10% interest compounded monthly. Then use that equation to fill the table and use the table to graph the equation.
### Solution
Given:
- Initial Investment (\( P \)) = $5000
- Annual Interest Rate (\( r \)) = 10% = 0.10
- Number of times interest is compounded per year (\( n \)) = 12 (monthly)
Substituting these values into the compound interest formula, we get:
\[ A = 5000 \left(1 + \frac{0.10}{12}\right)^{12t} \]
\[ A = 5000 \left(1 + \frac{0.10}{12}\right)^{12t} \]
\[ A = 5000 \left(1 + 0.00833\right)^{12t} \]
\[ A = 5000 (1.00833)^{12t} \]
### Fill in the Table
| Years (\( t \)) | Value (\( A \)) |
|-----------------|--------------------|
| 0 | $5000 |
| 5 | $ |
| 10 | $ |
| 15 | $ |
| 20 | $ |
To calculate the values, we plug in the corresponding \( t \) values into the equation:
For \( t = 5 \):
\[ A = 5000 (1.00833)^{12 \times 5} \]
For \( t = 10 \):
\[ A = 5000 (1.00833)^{12 \times 10} \]
For \( t = 15 \):
\[ A = 5000 (1.00833)^{12 \times 15} \]
For \( t = 20 \):
\[ A = 5000 (1.00833)^{12 \times
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