Problem 59 - College Algebra

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 five terms are 9, -1, -11, -21, and -31.

Step 1: Understanding the Arithmetic Sequence Formula

The arithmetic sequence formula is generally given as \(a_n = a_1 + (n-1)\cdot d \), where \(a_1\) is the first term and \(d\) is the common difference. The exercise provides \(a_1=9\) and the recursive formula \(a_n = a_{n-1} - 10\), indicating that the common difference \(d = -10\).

Step 2: Calculate the First Five Terms

Using the recursive formula, calculate the first five terms of the arithmetic sequence: \(a_1 = 9\), \(a_2 = a_1 - 10 = 9 - 10 = -1\), \(a_3 = a_2 - 10 = -1 - 10 = -11\), \(a_4 = a_3 - 10 = -11 - 10 = -21\), \(a_5 = a_4 - 10 = -21 - 10 = -31\). Thus, the terms are \(9, -1, -11, -21, -31\).

Step 3: Plot the Terms on a Graph

To graph, use a coordinate system where the x-axis represents the term number \(n\) and the y-axis represents the term value \(a_n\). Plot the points: \((1, 9), (2, -1), (3, -11), (4, -21), (5, -31)\). Connect the dots with a straight line, since the sequence is arithmetic.

Key Concepts

Arithmetic Sequence Formula

An arithmetic sequence is a list of numbers with a common difference between consecutive terms. It is defined by the formula \[ a_n = a_1 + (n-1) \cdot d \], where:

• \( a_n \) is the nth term,
• \( a_1 \) is the first term,
• \( n \) is the term number,
• \( d \) is the common difference.

This formula helps calculate any term in the sequence if you know the first term and the common difference.
In the given exercise, the first term \( a_1 \) is 9. The formula to find other terms in the sequence uses the recursive method, but it can be rewritten to fit the formula: \[ a_n = 9 + (n-1)(-10). \] This allows quick calculation of any term in the sequence.

Common Difference

The common difference \( d \) in an arithmetic sequence is the amount each term increases or decreases byas we move from one term to the next. It is found by subtracting any term from the term that follows it.
In the exercise, the recursive formula \( a_n = a_{n-1} - 10 \) shows that our common difference is -10. This means each term is 10 less than the previous one, creating a sequence which decreases.

• If the common difference is positive, the sequence is increasing.
• If the common difference is negative, like here, the sequence is decreasing.

The concept of common difference is crucial because it determines the trend and nature of the sequence.

Graphing Arithmetic Sequences

Graphing an arithmetic sequence involves using a coordinate system where:

• The x-axis represents the term number \( n \).
• The y-axis represents the value of the term \( a_n \).

For the exercise, the terms are: 9, -1, -11, -21, and -31. Each term corresponds to a point:

• \((1, 9)\)
• \((2, -1)\)
• \((3, -11)\)
• \((4, -21)\)
• \((5, -31)\)

After plotting these points, connect them with a straight line. This line shows the consistent decrease represented by the negative common difference. Arithmetic sequences create straight-line graphs because of the constant addition or subtraction in each step.

Recursive Formula

The recursive formula provides another way to define arithmetic sequences. It requires using the previous term to find the subsequent term. In our exercise, the recursive formula is \[ a_n = a_{n-1} - 10. \] This indicates that by subtracting 10 from the previous term, you can find the next term.
A key point about recursive formulas is their reliance on prior terms. You need to know at least one term to use them effectively. While they can be repetitive for larger sequences, they are highly intuitive and often easier to apply for shorter sequences. The recursive approach fits naturally with step-by-step processes, making it a practical choice for introductory exercises in sequences.