Eva opened a savings account with an initial deposit of $882. They then deposit $882 into that savings account at the end of every subsequent year. This savings account pays an annual interest rate of 3.1% and is compounded annually. How much will Eva have in their account after 6 years? In the image. Round your answer to the nearest penny.
Eva opened a savings account with an initial deposit of $882. They then deposit $882 into that savings account at the end of every subsequent year. This savings account pays an annual interest rate of 3.1% and is compounded annually. How much will Eva have in their account after 6 years? In the image. Round your answer to the nearest penny.
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Eva opened a savings account with an initial deposit of $882. They then deposit $882 into that savings account at the end of every subsequent year. This savings account pays an annual interest rate of 3.1% and is compounded annually. How much will Eva have in their account after 6 years? In the image. Round your answer to the nearest penny.
![The formula depicted in the image is used to calculate the balance of an annuity investment over time. Here is a detailed transcription and explanation suitable for an educational website:
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## Calculating Annuity Balance Over Time
The formula to determine the balance \( B(t) \) of an annuity investment at time \( t \) is:
\[ B(t) = P \cdot \left( \frac{\left(1 + \frac{r}{n}\right)^{n \cdot t} - 1}{\frac{r}{n}} \right) \]
### Explanation of the Variables:
- \( B(t) \): The balance of the annuity at time \( t \)
- \( P \): The payment amount per period
- \( r \): The annual interest rate (as a decimal)
- \( n \): The number of compounding periods per year
- \( t \): The number of years the money is invested or borrowed for
### Detailed Breakdown:
1. **Annual Interest Rate ( \( \frac{r}{n} \) )**:
- This term represents the interest rate per compounding period.
2. **Compounded Growth ( \( \left(1 + \frac{r}{n}\right)^{n \cdot t} \) )**:
- This component accounts for the compound growth of the investment over \( n \cdot t \) periods.
3. **Adjustment for Periodic Payments**:
- Subtracting 1 from the compounded growth factor adjusts for the initial starting value in order to isolate the growth attributable to the periodic payments.
4. **Overall Scaling**:
- Multiplying by \( P \) scales the entire formula to account for the actual payment amount per period.
- Dividing by \( \frac{r}{n} \) adjusts the result to represent the contribution of each payment relative to the interest rate per period.
### Summary:
This formula is useful for calculating the future value of an annuity investment considering regular deposits (\( P \)) made into an account that compounds interest \( n \) times per year at an annual interest rate \( r \) over \( t \) years.
Using this formula, investors can understand how their periodic investments will grow over time, allowing for better financial planning and goal setting.
---
Feel free to use this breakdown to enhance your understanding of how regular payments into an interest-bearing account compound over](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F27b4bc7f-06c3-4245-81d4-6188c9348506%2F31efe4f4-2aab-4941-a45b-251f945f0cbf%2F80ekxb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The formula depicted in the image is used to calculate the balance of an annuity investment over time. Here is a detailed transcription and explanation suitable for an educational website:
---
## Calculating Annuity Balance Over Time
The formula to determine the balance \( B(t) \) of an annuity investment at time \( t \) is:
\[ B(t) = P \cdot \left( \frac{\left(1 + \frac{r}{n}\right)^{n \cdot t} - 1}{\frac{r}{n}} \right) \]
### Explanation of the Variables:
- \( B(t) \): The balance of the annuity at time \( t \)
- \( P \): The payment amount per period
- \( r \): The annual interest rate (as a decimal)
- \( n \): The number of compounding periods per year
- \( t \): The number of years the money is invested or borrowed for
### Detailed Breakdown:
1. **Annual Interest Rate ( \( \frac{r}{n} \) )**:
- This term represents the interest rate per compounding period.
2. **Compounded Growth ( \( \left(1 + \frac{r}{n}\right)^{n \cdot t} \) )**:
- This component accounts for the compound growth of the investment over \( n \cdot t \) periods.
3. **Adjustment for Periodic Payments**:
- Subtracting 1 from the compounded growth factor adjusts for the initial starting value in order to isolate the growth attributable to the periodic payments.
4. **Overall Scaling**:
- Multiplying by \( P \) scales the entire formula to account for the actual payment amount per period.
- Dividing by \( \frac{r}{n} \) adjusts the result to represent the contribution of each payment relative to the interest rate per period.
### Summary:
This formula is useful for calculating the future value of an annuity investment considering regular deposits (\( P \)) made into an account that compounds interest \( n \) times per year at an annual interest rate \( r \) over \( t \) years.
Using this formula, investors can understand how their periodic investments will grow over time, allowing for better financial planning and goal setting.
---
Feel free to use this breakdown to enhance your understanding of how regular payments into an interest-bearing account compound over
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