Problem 27 - College Algebra with Corequisite Support

Question

For the following exercises, use the recursive formula to write the first five terms of the arithmetic sequence. $$ a_{1}=-19 ; \quad a_{n}=a_{n-1}-1.4 $$

Step-by-Step Solution

Answer

The first five terms are -19, -20.4, -21.8, -23.2, -24.6.

Step 1: Identify First Term

The first term of the sequence is given as \( a_1 = -19 \). This term is provided explicitly, so we can start from here.

Step 2: Understand Recursive Formula

The recursive formula \( a_n = a_{n-1} - 1.4 \) means that each term is obtained by subtracting 1.4 from the previous term.

Step 3: Calculate Second Term

Using the recursive formula: \( a_2 = a_1 - 1.4 = -19 - 1.4 = -20.4 \).

Step 4: Calculate Third Term

Continuing the pattern: \( a_3 = a_2 - 1.4 = -20.4 - 1.4 = -21.8 \).

Step 5: Calculate Fourth Term

Apply the formula again: \( a_4 = a_3 - 1.4 = -21.8 - 1.4 = -23.2 \).

Step 6: Calculate Fifth Term

Finally, calculate the fifth term: \( a_5 = a_4 - 1.4 = -23.2 - 1.4 = -24.6 \).

Step 7: Summarize First Five Terms

The first five terms of the sequence are: \(-19, -20.4, -21.8, -23.2, -24.6\).

Key Concepts

Understanding Recursive Formulas

A recursive formula is a method used to define the terms in a sequence with respect to the preceding terms. In arithmetic sequences, a recursive formula helps to find any term based on the one before it. For example, consider the sequence given by the formula \( a_n = a_{n-1} - 1.4 \). This implies:

• The sequence starts at a specific number or the first term, \( a_1 \).
• Each subsequent term, \( a_n \), is derived by subtracting 1.4 from the term immediately before it, \( a_{n-1} \).

Recursive formulas are particularly useful because they provide a clear, step-by-step process to build the entire sequence. The biggest advantage of using a recursive formula is that you don't need to know the "pattern" or explicit formula upfront; you can generate terms directly following the given instructions.

Identifying Sequence Terms

Terms of a sequence are individual elements or numbers arranged in a specific order. In the arithmetic sequence exercise provided, the terms use a starting value and a rule to form the entire pattern. Let's break down what we've done:

• The first term \( a_1 = -19 \) is given, setting the sequence's starting point.
• The second term \( a_2 = -20.4 \) is calculated using the recursive rule: \( a_2 = a_1 - 1.4 = -19 - 1.4 \).
• Subsequent terms continue in this fashion: \( a_3 = a_2 - 1.4 \), \( a_4 = a_3 - 1.4 \), and so forth.

This systematic approach ensures clarity and consistency, allowing you to generate sequences over many terms without confusion. Mastery of sequence terms helps solidify understanding of how each part of a sequence connects to the others.

Solving Algebra Problems with Sequences

Arithmetic sequences form a foundational part of algebra, providing a practical application for solving real-life problems. When tackling algebra problems involving sequences:

• Identify patterns using the recursive formula. The formula gives a repeatable method to address each part of the sequence.
• Understand the impact of the common difference. In the example \( a_n = a_{n-1} - 1.4 \), the number -1.4 represents how each term changes relative to its predecessor.
• Implement algebraic operations systematically to adjust the sequence as required by the problem.

Arithmetic sequences can appear in various real-world scenarios, such as calculating interest, predicting growth, or organizing schedules. Therefore, grasping sequence operations helps enhance problem-solving skills in broader mathematical contexts. Algebras problems often build upon these basic constructs to explore more complex concepts.