Problem 59 - College Algebra with Corequisite Support

Question

For the following exercises, use the information provided to graph the first 5 terms of the arithmetic sequence. \(a_{1}=9 ; a_{n}=a_{n-1}-10\)

Step-by-Step Solution

Answer

The first 5 terms are: 9, -1, -11, -21, -31.

Step 1: Identify the Initial Term

The first term of the sequence, denoted by \(a_1\), is given as 9. This is the starting point of the sequence.

Step 2: Understand the Recursive Formula

The recursive formula for the sequence is given as \(a_n = a_{n-1} - 10\). This means that each subsequent term is 10 less than the previous term.

Step 3: Calculate the First 5 Terms

Using the initial term and the recursive formula, calculate the first 5 terms:1. \(a_1 = 9\)2. \(a_2 = a_1 - 10 = 9 - 10 = -1\)3. \(a_3 = a_2 - 10 = -1 - 10 = -11\)4. \(a_4 = a_3 - 10 = -11 - 10 = -21\)5. \(a_5 = a_4 - 10 = -21 - 10 = -31\)

Step 4: Plot the Terms on a Graph

Graph these calculated terms on a coordinate plane. The x-axis represents the term number \(n\), and the y-axis represents the value of each term \(a_n\). Plot the points (1, 9), (2, -1), (3, -11), (4, -21), and (5, -31).

Key Concepts

Recursive Formula

A recursive formula is a way to define a sequence where each term is determined by one or more previous terms. In mathematics, this formula is especially useful for arithmetic and other sequences. For this particular sequence, the recursive formula is given as \( a_n = a_{n-1} - 10 \). This means that to find the value of any term in the sequence, you subtract 10 from the term before it.

• Start with an initial term, which is provided in this exercise as \( a_1 = 9 \).
• Use the recursive formula to derive the next terms by repeatedly subtracting 10.

This approach creates a rule you can follow to generate additional terms of the sequence, which is both systematic and efficient.

Graphing Sequences

Plotting sequences on a graph can provide a visual representation that helps in understanding how a sequence progresses. When graphing an arithmetic sequence, each term can be considered a point on the plane. For this sequence, each term you've calculated is a point: for example, \((1,9)\) is the first term plotted on the graph.

• The x-axis indicates the position of the term within the sequence, often represented by \(n\).
• The y-axis represents the term's value— here, it’s the calculated \(a_n\) based on the recursive formula.

You should notice that, for an arithmetic sequence like this one, the plotted points form a straight line. This linearity is a distinctive characteristic of arithmetic sequences and helps verify if the sequence is correctly represented.

Coordinate Plane

When it comes to arithmetic sequences, the coordinate plane is a crucial tool for visualizing patterns and relationships between the terms. A coordinate plane consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).

• The x-axis typically represents the sequence's position number (n).
• The y-axis represents each term's value in the sequence (\(a_n\)).

For our particular case, you can plot the points (1, 9), (2, -1), (3, -11), (4, -21), and (5, -31) on the plane, which clearly illustrates the sequence's progression and gives a visual sense of its direction and rate of change.

Sequence Calculations

Sequence calculations involve determining the values of different terms within a sequence using a specified rule. For arithmetic sequences, these calculations are straightforward due to the constant difference between consecutive terms. Building upon our initial step-by-step solution, you calculated the first five terms using repetition and the recursive formula:

• Compute \(a_1\) from the given initial term.
• Apply the recursive rule iteratively: subtract 10 from the last calculated term to find the next.
• Continue this method to achieve the desired number of terms.

This method ensures accuracy while following the sequence’s specific pattern. Arithmetic computations like these help in understanding the deeper relationships that govern the sequence.