Problem 42 - College Algebra

Question

For the following exercises, use the recursive formula to write the first five terms of the arithmetic sequence. $$ a_{n}=\frac{1}{2} n-\frac{1}{2} $$

Step-by-Step Solution

Answer

The first five terms are 0, \(\frac{1}{2}\), 1, \(\frac{3}{2}\), and 2.

Step 1: Understand the Recursive Formula

The given recursive formula for the arithmetic sequence is \( a_n = \frac{1}{2}n - \frac{1}{2} \). This formula will be used to calculate the sequence values for the first five terms \( a_1, a_2, a_3, a_4, \) and \( a_5 \).

Step 2: Calculate the First Term \( a_1 \)

Substitute \( n = 1 \) into the formula: \( a_1 = \frac{1}{2}(1) - \frac{1}{2} = \frac{1}{2} - \frac{1}{2} = 0 \).

Step 3: Calculate the Second Term \( a_2 \)

Substitute \( n = 2 \) into the formula: \( a_2 = \frac{1}{2}(2) - \frac{1}{2} = 1 - \frac{1}{2} = \frac{1}{2} \).

Step 4: Calculate the Third Term \( a_3 \)

Substitute \( n = 3 \) into the formula: \( a_3 = \frac{1}{2}(3) - \frac{1}{2} = \frac{3}{2} - \frac{1}{2} = 1 \).

Step 5: Calculate the Fourth Term \( a_4 \)

Substitute \( n = 4 \) into the formula: \( a_4 = \frac{1}{2}(4) - \frac{1}{2} = 2 - \frac{1}{2} = \frac{3}{2} \).

Step 6: Calculate the Fifth Term \( a_5 \)

Substitute \( n = 5 \) into the formula: \( a_5 = \frac{1}{2}(5) - \frac{1}{2} = \frac{5}{2} - \frac{1}{2} = 2 \).

Key Concepts

Understanding Recursive Formulas

A recursive formula is a powerful tool used in mathematics to generate the terms of a sequence. In the case of an arithmetic sequence, each term is derived by following a specific pattern or rule. This type of formula is particularly useful because, rather than recalculating everything from scratch, you simply use the previous term to find the next one.

For the arithmetic sequence in this exercise, the recursive formula is given by \( a_n = \frac{1}{2}n - \frac{1}{2} \). This tells us exactly how to calculate each term one after the other.

It's important to note that the recursive formula reflects the common difference, which is the fixed number added to each term to get to the next. With this formula, we're able to map out the sequence:

• First, identify \( n \), the term number in the sequence.
• Secondly, substitute this number into the formula to find the term \( a_n \).

This structured approach simplifies finding multiple terms in a sequence without repetitive calculations.

Identifying Sequence Terms

Sequence terms are the individual elements or numbers that make up a sequence. In an arithmetic sequence, these terms have a constant difference between consecutive terms.

In our example, terms are derived using a specific order: the first term \( a_1 \), the second term \( a_2 \), and so on.
Sequence terms are invaluable in understanding the behavior of the entire sequence as they represent the actual numbers generated by the recursive formula.

• For \( a_1 \), substitute \( n = 1 \) into the formula to get \( a_1 = 0 \).
• For \( a_2 \), substitute \( n = 2 \) to find \( a_2 = \frac{1}{2} \).
• Continue this process to calculate terms like \( a_3 \), \( a_4 \), and \( a_5 \).

By understanding how the sequence terms are derived, you begin to see the sequence's pattern or trend, which can help predict or understand future or previous terms.

The Role of Algebra in Sequences

Algebra plays a pivotal role in managing sequences, particularly when using formulas and expressions. It's the foundation that allows us to translate the logic of sequences into mathematical language. In our recursive formula \( a_n = \frac{1}{2}n - \frac{1}{2} \), algebra helps us understand how changes in \( n \) influence the sequence terms.

Here's how algebra is applied effectively:

• Substitution: By replacing the variable \( n \) with different integers, we apply algebraic operations to arrive at the sequence terms.
• Simplification: Algebra allows us to simplify our expressions, making it easier to compute the terms efficiently.
• Pattern recognition: Algebra helps in identifying the arithmetic foundation, and therefore, the common difference within sequences.

These techniques make it possible to not just compute sequence terms but also to explore deeper relationships between them, enhancing our problem-solving capabilities.

Effective Problem Solving Techniques in Sequences

Problem-solving in sequences involves strategic thinking and systematic exploration of the rules governing the sequence. To effectively tackle arithmetic sequence problems, one should:

• Understand the problem: Carefully interpret the information provided, paying particular attention to the formula or rule that defines the sequence.
• Follow steps: Start by computing the sequence terms step-by-step, ensuring that logical operations follow algebraic guidelines.
• Review and verify: Once the terms are calculated, double-check your steps and results against the formula to confirm accuracy.

Effective problem-solving not only involves brute calculation but also a consistent approach and verification. Understanding the sequence's underlying pattern aids in foreseeing potential errors, ensuring solutions are reliable. By practicing these techniques, one gains confidence and skill in handling increasingly complex sequence problems.