Martina Opens A Savings Account With An Initial Deposit And Makes No Other Deposits Or Withdrawals. She Earns Interest On Her Initial Deposit. The Total Amount Of Money In Her Savings Account At The End Of Each Year Is Represented By The Sequence
Introduction
In this article, we will explore a savings account scenario where Martina makes an initial deposit and earns interest on it. The total amount of money in her savings account at the end of each year forms a sequence. We will analyze this sequence and understand its properties.
The Savings Account Sequence
Let's assume Martina opens a savings account with an initial deposit of $1000. She earns an annual interest rate of 5%. The total amount of money in her savings account at the end of each year is represented by the sequence:
1000, 1050, 1102.50, 1157.63, 1215.09, ...
This sequence represents the total amount of money in Martina's savings account at the end of each year.
Geometric Sequence
The savings account sequence is a geometric sequence. A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
In this case, the common ratio is 1.05, which represents the annual interest rate of 5%. The first term of the sequence is 1000, and each subsequent term is found by multiplying the previous term by 1.05.
Formula for the Savings Account Sequence
The formula for a geometric sequence is:
an = ar^(n-1)
where:
- an is the nth term of the sequence
- a is the first term of the sequence (1000 in this case)
- r is the common ratio (1.05 in this case)
- n is the term number (1, 2, 3, ...)
Using this formula, we can calculate the total amount of money in Martina's savings account at the end of each year.
Calculating the Savings Account Balance
Let's calculate the total amount of money in Martina's savings account at the end of each year using the formula:
| Year | Term Number (n) | Savings Account Balance (an) |
|---|---|---|
| 1 | 1 | 1000 |
| 2 | 2 | 1000 × 1.05 = 1050 |
| 3 | 3 | 1050 × 1.05 = 1102.50 |
| 4 | 4 | 1102.50 × 1.05 = 1157.63 |
| 5 | 5 | 1157.63 × 1.05 = 1215.09 |
As we can see, the savings account balance increases by 5% each year, resulting in a geometric sequence.
Savings Account Balance after n Years
To find the savings account balance after n years, we can use the formula:
an = 1000 × (1.05)^(n-1)
This formula allows us to calculate the total amount of money in Martina's savings account at the end of any year.
Example Use Case
Let's say Martina wants to know the savings account balance after 10 years. We can use the formula:
an = 1000 × (1.05)^(10-1) = 1000 × (1.05)^9 = 1000 × 1.6289 = 1628.90
Therefore, the savings account balance after 10 years is $1628.90.
Conclusion
In this article, we analyzed the savings account sequence and understood its properties. We found that the sequence is a geometric sequence with a common ratio of 1.05. We also derived a formula for the savings account balance after n years and used it to calculate the balance after 10 years. This formula can be used to calculate the savings account balance after any number of years.
References
- [1] "Geometric Sequence." Math Open Reference, mathopenref.com/sequences-geometric.html.
- [2] "Savings Account Balance." Investopedia, investopedia.com/terms/s/savings-account-balance.asp.
Further Reading
- "Geometric Sequences and Series." Khan Academy, khanacademy.org/math/sequences-series/sequences-series-review-article/sequences-series-review-article#geometric-sequences.
- "Savings Account Interest Rates." Bankrate, bankrate.com/banking/savings-account-interest-rates/.
Frequently Asked Questions (FAQs) about the Savings Account Sequence ====================================================================
Q: What is the savings account sequence?
A: The savings account sequence is a sequence of numbers that represents the total amount of money in a savings account at the end of each year. It is a geometric sequence, where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
Q: What is the common ratio in the savings account sequence?
A: The common ratio in the savings account sequence is 1.05, which represents the annual interest rate of 5%.
Q: How do I calculate the savings account balance after n years?
A: To calculate the savings account balance after n years, you can use the formula:
an = 1000 × (1.05)^(n-1)
where:
- an is the nth term of the sequence
- 1000 is the initial deposit
- 1.05 is the common ratio
- n is the term number (1, 2, 3, ...)
Q: What is the savings account balance after 10 years?
A: To find the savings account balance after 10 years, we can use the formula:
an = 1000 × (1.05)^(10-1) = 1000 × (1.05)^9 = 1000 × 1.6289 = 1628.90
Therefore, the savings account balance after 10 years is $1628.90.
Q: How does the savings account sequence change if the interest rate changes?
A: If the interest rate changes, the common ratio in the savings account sequence will also change. For example, if the interest rate is 6%, the common ratio will be 1.06. To calculate the savings account balance after n years with a different interest rate, you can use the formula:
an = 1000 × (1.06)^(n-1)
Q: Can I use the savings account sequence to calculate the total amount of money in a savings account after a certain number of years with a different initial deposit?
A: Yes, you can use the savings account sequence to calculate the total amount of money in a savings account after a certain number of years with a different initial deposit. To do this, you can modify the formula to include the new initial deposit:
an = P × (1.05)^(n-1)
where:
- an is the nth term of the sequence
- P is the new initial deposit
- 1.05 is the common ratio
- n is the term number (1, 2, 3, ...)
Q: What are some real-world applications of the savings account sequence?
A: The savings account sequence has many real-world applications, including:
- Calculating the total amount of money in a savings account after a certain number of years
- Determining the interest rate required to reach a certain savings goal
- Comparing the performance of different savings accounts with different interest rates
- Calculating the future value of a series of deposits
Q: How can I use the savings account sequence in my personal finance planning?
A: You can use the savings account sequence to plan your personal finances by:
- Calculating the total amount of money you will have in your savings account after a certain number of years
- Determining the interest rate required to reach your savings goal
- Comparing the performance of different savings accounts with different interest rates
- Calculating the future value of a series of deposits
Conclusion
In this article, we answered some frequently asked questions about the savings account sequence. We covered topics such as the common ratio, calculating the savings account balance after n years, and real-world applications of the savings account sequence. We hope this article has been helpful in understanding the savings account sequence and how it can be used in personal finance planning.