Problem 5 - College Algebra

Question

In Exercises \(1-14\), write the first six terms of cach arithmetic sequence $$a_{1}=300, d=-90$$

Step-by-Step Solution

Answer

The first six terms of the arithmetic sequence are 300, 210, 120, 30, -60, -150 respectively.

Step 1: Identify Initial Term and Common Difference

The initial term, denoted by \(a_1\), is 300 and the common difference, denoted by 'd', is -90. These have been provided in the question.

Step 2: Use the Formula to find the remaining terms

Following the formula for an arithmetic sequence, we compute the terms using \(a_n = a_1 + (n-1) * d\). Substituting \(a_1=\)300 and \(d=-90\) in the formula, we get the first six terms by substituting n=1,2,3,4,5,6 sequentially.

Step 3: Calculate the first six terms

For n=1, we find \(a_1= 300\). For n=2, we find \(a_2= 300 + (2-1)*{-90} = 210\). For n=3, we find \(a_3= 300+(3-1)*{-90} =120\). For n=4, we find \(a_4 = 300 + (4-1)*{-90} =30\). For n=5, we find \(a_5= 300+(5-1)*{-90}= -60\). Finally, for n=6, we find \(a_6= 300+(6-1)*{-90}= -150\).

Key Concepts

Initial Term

An arithmetic sequence is defined by its initial term and common difference. The **initial term** is the starting point of the sequence, denoted usually as \( a_1 \). In this particular exercise, the initial term is given as \( a_1 = 300 \).
This value represents the very first number in the sequence.
Understanding the initial term is crucial because it sets the baseline for the rest of the sequence.

• Position in Sequence: First term, \( a_1 \).
• Role: It is not influenced by a common difference directly.
• Importance: Determines the absolute position of all following terms.

Common Difference

The arithmetic sequence is characterized by a **common difference** between its terms. This means each successive term is obtained by adding or subtracting this specific value from the previous term.
In our exercise, the common difference is \(-90\).
This negative common difference indicates that each term is \(90\) less than the previous one.

• Symbol: Denoted by \(d\).
• Value: Can be positive, negative, or zero.
• Effect: Affects the sequence's direction and spacing between terms.

Sequential Terms

To list an arithmetic sequence, we apply both the initial term and the common difference.
The sequential terms result from repeatedly adding the common difference to the previous term. Following this process helps to construct the series step-by-step, ensuring accuracy. In our example sequence, derived from \(a_1 = 300\) and \(d = -90\), the terms are:

• \(a_1 = 300\)
• \(a_2 = 210\)
• \(a_3 = 120\)
• \(a_4 = 30\)
• \(a_5 = -60\)
• \(a_6 = -150\)

These values illustrate how the sequence progresses continuously at the same rate.

Formula for Arithmetic Sequence

One of the core aspects of understanding arithmetic sequences is using their formula. The **formula** is used to find any term within the sequence, defined as \( a_n = a_1 + (n-1) \times d \).
This formula considers:

• \(a_n\): any term in the sequence, based on its position, \(n\).
• \(a_1\): the known initial term of the sequence.
• \(d\): the common difference.

The formula is a powerful tool that allows the calculation of any term, not just the first few, offering flexibility and ease in dealing with long sequences.
It shows the dependence of sequence terms on both a consistent difference and the initial value.