Problem 26
Question
For the following exercises, use the following scenario. Javier makes monthly deposits into a savings account. He opened the account with an initial deposit of \(\$ 50\). Each month thereafter he increased the previous deposit amount by \(\$ 20\). Graph the arithmetic sequence showing one year of Javier's deposits.
Step-by-Step Solution
Verified Answer
Javier's deposits over a year are: 50, 70, 90, 110, 130, 150, 170, 190, 210, 230, 250, 270. Graph these as points on a coordinate plane.
1Step 1: Understand the Sequence
Javier's deposits increase by a constant amount each month. This is an arithmetic sequence where the first term is the initial deposit of $50, and the common difference is $20.
2Step 2: Determine the Sequence Terms
Each month, the deposit amount increases by \(20. If the initial deposit is \)50, the sequence for one year (12 months) is: \(a_1 = 50, a_2 = 70, a_3 = 90, \dots, a_{12}\). The general formula is \(a_n = a_1 + (n-1) \times d\), where \(d=20\).
3Step 3: Calculate Each Month's Deposit
Using the formula \(a_n = 50 + (n-1) \times 20\), find each term for \(n = 1, 2, \ldots, 12\). The terms are: \(50, 70, 90, 110, 130, 150, 170, 190, 210, 230, 250, 270\).
4Step 4: Graph the Sequence
On a coordinate plane, the x-axis represents the month, and the y-axis represents the deposit amount. Plot the points: \((1, 50), (2, 70), (3, 90), \ldots, (12, 270)\). Each point connects to form a line that reflects the arithmetic increase in deposits.
Key Concepts
Initial DepositCommon DifferenceGraph SequenceCoordinate Plane
Initial Deposit
The initial deposit is the amount Javier placed into his savings account when he first opened it. In this context, Javier made an initial deposit of $50. This amount is crucial because it serves as the starting point for the arithmetic sequence, from which all subsequent deposit amounts are calculated.
The concept of an initial deposit is simple yet foundational in financial planning and arithmetic sequences. It is essentially the first term in our sequence, often denoted as \(a_1\).
In arithmetic sequences, especially involving financial growth like saving plans or investments, this initial amount sets the pace for how the growth unfolds over time.
The concept of an initial deposit is simple yet foundational in financial planning and arithmetic sequences. It is essentially the first term in our sequence, often denoted as \(a_1\).
In arithmetic sequences, especially involving financial growth like saving plans or investments, this initial amount sets the pace for how the growth unfolds over time.
Common Difference
The common difference in an arithmetic sequence is the constant amount added to each term to get the next term. For Javier, the common difference is \(20. This means every month, Javier deposits \)20 more than he did the previous month.
The common difference is a key element that defines the progression of the sequence. Mathematically, it is represented by \(d\).
Understanding this helps clarify how quickly or slowly a sequence grows. In Javier's case, a consistent increase of $20 each month shows a steady and predictable growth pattern in his savings.
The common difference is a key element that defines the progression of the sequence. Mathematically, it is represented by \(d\).
Understanding this helps clarify how quickly or slowly a sequence grows. In Javier's case, a consistent increase of $20 each month shows a steady and predictable growth pattern in his savings.
Graph Sequence
Graphing the arithmetic sequence of Javier's deposit amounts helps visualize the changes over time. By plotting each month's deposit on a coordinate plane, you can see how the deposits increase steadily. Each plotted point represents a specific month's deposit amount.
Seeing the graph provides an immediate visual representation of the arithmetic sequence and its consistent growth pattern. This makes it easier to understand how the deposits evolve over the year.
- On the x-axis, label each month (1 to 12).
- On the y-axis, label the deposit amounts, ranging from $50 to $270.
Seeing the graph provides an immediate visual representation of the arithmetic sequence and its consistent growth pattern. This makes it easier to understand how the deposits evolve over the year.
Coordinate Plane
A coordinate plane is a two-dimensional surface where you can plot points, lines, and curves to represent mathematical concepts visually. It's made up of two axes: the x-axis (horizontal) and the y-axis (vertical).
In Javier's savings example, each month's deposit is plotted on this plane, with months on the x-axis and deposit amounts on the y-axis.
In Javier's savings example, each month's deposit is plotted on this plane, with months on the x-axis and deposit amounts on the y-axis.
- The x-axis represents time or sequence order (e.g., each month).
- The y-axis corresponds to the value of the deposit amount.
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