Savings Account Growth: Arithmetic Sequence For $180 Deposit
Let's explore how Imogen's savings account grows with simple interest. With an initial deposit of $180 from her summer job and an interest rate of 4.5%, we can model her account balance each year using an arithmetic sequence. This article will guide you through calculating the interest, understanding the arithmetic sequence formula, and determining the correct representation of Imogen's savings growth.
Calculating Simple Interest and Its Impact
To understand the arithmetic sequence, we first need to calculate the simple interest Imogen earns each year. Simple interest is calculated only on the principal amount, which in this case is $180. The formula for simple interest is:
Interest = Principal × Rate × Time
Here, the principal is $180, the interest rate is 4.5% (or 0.045 as a decimal), and the time is 1 year. Plugging these values into the formula:
Interest = $180 × 0.045 × 1 = $8.10
This means Imogen earns $8.10 in interest each year. Since it's simple interest, this amount remains constant every year. This constant increase is the key to understanding the arithmetic sequence that represents her savings growth.
Now, let's delve into what an arithmetic sequence is and how it applies to Imogen's situation. An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is known as the common difference. In the context of Imogen's savings account, the common difference is the annual interest she earns, which we calculated to be $8.10. The general form of an arithmetic sequence is:
a_n = a_1 + (n - 1)d
Where:
- a_n is the nth term in the sequence (the account balance in the nth year)
- a_1 is the first term (the initial deposit)
- n is the term number (the year)
- d is the common difference (the annual interest)
In Imogen's case:
- a_1 = $180 (her initial deposit)
- d = $8.10 (the annual interest)
Substituting these values into the arithmetic sequence formula, we get:
a_n = $180 + (n - 1)$8.10
This formula represents Imogen's account balance each year. It starts with her initial deposit of $180 and adds $8.10 for each year that passes. Understanding this formula is crucial for identifying the correct arithmetic sequence representation among the given options.
Evaluating the Arithmetic Sequence Options
Now that we've established the formula for Imogen's savings growth, let's compare it to the provided options to determine the correct one. The options given are:
A. B.
Our derived formula, based on the simple interest calculation, is:
a_n = $180 + (n - 1)$8.10
Comparing the options, we can see that Option A exactly matches our derived formula. It correctly represents the initial deposit of $180 and the annual interest of $8.10 as the common difference.
Option B, on the other hand, is incorrect. It uses the interest rate (4.5) as the initial term and the initial deposit ($180) in the common difference part of the formula. This does not accurately represent how simple interest works. Simple interest is calculated on the principal, and the interest earned is added to the principal each year. Option B misinterprets these roles, leading to an incorrect representation of the savings growth.
Therefore, the correct arithmetic sequence that represents Imogen's account balance each year is Option A. This option accurately reflects the starting amount and the constant annual interest earned, providing a clear and precise model of her savings growth over time.
Understanding the Implications of the Arithmetic Sequence
Having identified the correct arithmetic sequence, it’s beneficial to understand what this sequence tells us about Imogen's savings. The sequence allows us to predict her account balance for any given year. For example, let's calculate her balance after 5 years:
a_5 = 180 + (5 - 1)8.10 a_5 = 180 + (4)8.10 a_5 = 180 + 32.40 a_5 = $212.40
After 5 years, Imogen's account balance will be $212.40. This demonstrates the power of arithmetic sequences in modeling situations with constant additive growth. It's a straightforward way to project future values based on a consistent rate of increase.
Furthermore, understanding this arithmetic sequence helps in financial planning. Imogen can use this model to see how her savings will grow over time with the current interest rate. If she has a savings goal in mind, she can adjust her initial deposit or look for accounts with higher interest rates to reach her goal faster. The arithmetic sequence provides a clear framework for visualizing and planning her savings journey.
In conclusion, the arithmetic sequence accurately represents Imogen's savings account balance each year, given her initial deposit of $180 and a simple interest rate of 4.5%. This understanding not only solves the immediate problem but also provides a valuable tool for financial planning and forecasting.
To pinpoint the arithmetic sequence that accurately depicts Imogen's savings account balance annually, we need to break down the components of simple interest and how they translate into an arithmetic sequence. The core of this problem lies in understanding how simple interest affects the growth of the principal amount over time.
First, let's revisit the concept of simple interest. As mentioned earlier, simple interest is calculated solely on the principal amount. This means that each year, the interest earned remains constant. This is a crucial distinction from compound interest, where the interest earned also earns interest in subsequent periods. The formula for simple interest is:
Interest = Principal × Rate × Time
In Imogen's situation, the principal is $180, the annual interest rate is 4.5% (or 0.045), and the time is measured in years. Calculating the annual interest earned:
Annual Interest = $180 × 0.045 × 1 = $8.10
This $8.10 is the constant amount that will be added to Imogen's account each year. This constant addition is what forms the basis of the arithmetic sequence. An arithmetic sequence, as we've discussed, is a sequence where the difference between consecutive terms is constant. This constant difference is known as the common difference.
Now, let's define the elements of the arithmetic sequence in the context of Imogen's savings:
- The first term, a_1, is the initial deposit, which is $180.
- The common difference, d, is the annual interest earned, which is $8.10.
Using the general formula for an arithmetic sequence:
a_n = a_1 + (n - 1)d
We can substitute Imogen's values:
a_n = $180 + (n - 1)$8.10
This formula allows us to calculate Imogen's account balance (a_n) for any given year (n). For instance, in the first year (n = 1), her balance is $180. In the second year (n = 2), it's $180 + $8.10 = $188.10, and so on.
Now, let's re-examine the options provided:
A. B.
Comparing these options with our derived formula, it's clear that Option A is the correct representation. It accurately captures the initial deposit of $180 and the consistent annual interest of $8.10. Option B, however, incorrectly places the interest rate as the initial term and the initial deposit as part of the common difference, which doesn't align with the principles of simple interest calculation.
Therefore, the arithmetic sequence that accurately represents Imogen's savings account balance each year is:
a_n = $180 + (n - 1)$8.1
This sequence starts at $180 and increases by $8.10 each year, reflecting the constant interest earned under a simple interest model. Understanding this sequence is crucial for both solving the problem and grasping the fundamentals of financial growth with simple interest.
Applying the Arithmetic Sequence in Real-World Scenarios
Beyond the specific problem of Imogen's savings account, understanding arithmetic sequences has broader applications in financial planning and other real-world scenarios. Let's explore some ways in which this concept can be applied.
One of the most significant applications is in financial forecasting. By using an arithmetic sequence, individuals can project how their savings or investments will grow over time, assuming a consistent rate of return. While simple interest is a basic model, the concept can be extended to scenarios with more complex interest calculations or varying rates of return. For instance, if someone is saving a fixed amount each month, this can also be modeled as an arithmetic sequence, with the monthly contribution as the common difference.
In addition to personal finance, arithmetic sequences are used in business and economics to model linear growth patterns. For example, a company's sales might increase by a fixed amount each quarter, or a population might grow at a constant rate. In these cases, an arithmetic sequence can provide a useful tool for predicting future values and making informed decisions.
Another application is in calculating loan repayments. While many loans use compound interest, understanding arithmetic sequences can help in grasping the basic principles of amortization. For instance, the portion of a loan payment that goes towards the principal can sometimes follow an arithmetic sequence, especially in the early years of the loan.
Furthermore, arithmetic sequences have applications outside of finance. They can be used in physics to model motion with constant acceleration, in computer science to analyze algorithms, and in various other fields where quantities increase or decrease at a steady rate. The fundamental principle of a constant difference between terms makes arithmetic sequences a versatile tool for modeling linear growth in a wide range of contexts.
In conclusion, the ability to recognize and apply arithmetic sequences is a valuable skill. Whether it's for understanding personal savings growth, forecasting business trends, or modeling physical phenomena, arithmetic sequences provide a simple yet powerful framework for analyzing situations with linear growth patterns. Understanding Imogen's savings account through an arithmetic sequence is just one example of the many ways this concept can be used in practice.
Conclusion
In summary, by calculating the simple interest earned annually and applying the arithmetic sequence formula, we've determined that the correct representation of Imogen's savings account balance each year is:
A.
This understanding not only solves the problem but also highlights the practical applications of arithmetic sequences in financial planning and forecasting. For further information on simple interest and arithmetic sequences, consider exploring resources like Khan Academy's lessons on arithmetic sequences and series.
