Problem 14 - College Algebra

Question

In Exercises \(1-14\), write the first six terms of cach arithmetic sequence $$a_{n}=a_{n-1}-03, a_{1}=-1.7$$

Step-by-Step Solution

Answer

The first six terms of the arithmetic sequence are: -1.7, -2.0, -2.3, -2.6, -2.9, -3.2.

Step 1: Identify the first term

Identify the first term in the sequence. Here, it is given as \(a_1 = -1.7\), so the first term is -1.7.

Step 2: Apply the arithmetic sequence formula for second term

Apply the formula for the nth term of an arithmetic sequence to find the second term. In this scenario it would look like this: \(a_2 = a_{2-1} - 0.3 = a_1 - 0.3 = -1.7 - 0.3 = -2.0\). Therefore, the second term in the sequence is -2.0.

Step 3: Continue to apply the arithmetic sequence formula to find the remaining terms

Apply the formula to find third to sixth term: \n\n\(a_3 = a_{2} - 0.3 = -2.0 - 0.3 = -2.3\)\n\n\(a_4 = a_{3} - 0.3 = -2.3 - 0.3 = -2.6\)\n\n\(a_5 = a_{4} - 0.3 = -2.6 - 0.3 = -2.9\)\n\n\(a_6 = a_{5} - 0.3 = -2.9 - 0.3 = -3.2\)\n\nSo, the third term is -2.3, the fourth term is -2.6, the fifth term is -2.9 and the sixth term is -3.2.

Key Concepts

Term Identification in Arithmetic Sequences

Term identification is the foundation when dealing with arithmetic sequences. The first term, often designated as \(a_1\), sets the starting point of the sequence. In this exercise, the first term is \(-1.7\). This is crucial, as it serves as the base from which all other terms are derived. Once identified, each subsequent term depends on this initial value.
By understanding that this first term is \(-1.7\), you can begin to predict the pattern of the sequence. An important aspect of this is recognizing the common difference, which in this case is \(-0.3\).

• Check the given first term \(a_1\).
• Identify the common difference \(d\) in the formula.
• Use these values to find additional terms.

Through correctly identifying the initial term and the common difference, you create a roadmap for the rest of the sequence.

Understanding the Sequence Formula

In arithmetic sequences, the nth term formula enables you to determine any term given its position. The formula typically is written as \(a_n = a_{n-1} + d\). However, in this problem, the formula is expressed as \(a_n = a_{n-1} - 0.3\).
Here, each term is calculated by subtracting the common difference from the previous term. The sequence essentially moves in steady steps, either ascending or descending, based only on the common difference.

• Start with the first term \(a_1\).
• Apply the common difference repeatedly to find subsequent terms.
• Notice the consistent change in the sequence defined by \(d\).

This simple but powerful tool helps you find any term by understanding its position within the sequence without having to compute every preceding term explicitly.

Step-by-Step Solution to Find Terms

Walking through the process of finding each term step-by-step helps solidify understanding of arithmetic sequences. We've established that \(a_1 = -1.7\) and the common difference is \(-0.3\). Now, let's follow the steps to find the first six terms.
First, the second term: \(a_2 = a_{1} - 0.3 = -1.7 - 0.3 = -2.0\). Now you have the second term.
Repeat this process:

• Third term: \(a_3 = a_{2} - 0.3 = -2.0 - 0.3 = -2.3\)
• Fourth term: \(a_4 = a_{3} - 0.3 = -2.3 - 0.3 = -2.6\)
• Fifth term: \(a_5 = a_{4} - 0.3 = -2.6 - 0.3 = -2.9\)
• Sixth term: \(a_6 = a_{5} - 0.3 = -2.9 - 0.3 = -3.2\)

Follow these steps to efficiently calculate each term, ensuring accuracy and reinforcing your understanding of the arithmetic sequence's behavior.