Problem 36 - College Algebra with Corequisite Support

Question

For the following exercises, write a recursive formula for each arithmetic sequence. $$ a=\left\\{-\frac{1}{2},-\frac{5}{4},-2, \ldots\right\\} $$

Step-by-Step Solution

Answer

The recursive formula is \(a_n = a_{n-1} - \frac{3}{4}\) with \(a_1 = -\frac{1}{2}\).

Step 1: Identify the Common Difference

The first step in working with an arithmetic sequence is to find the common difference between consecutive terms. For the sequence \(a = \{-\frac{1}{2}, -\frac{5}{4}, -2, \ldots\}\), calculate the difference between the second term and the first term: \(-\frac{5}{4} - \left(-\frac{1}{2}\right)\). Simplifying this, we get: \(-\frac{5}{4} + \frac{2}{4} = -\frac{3}{4}\). Thus, the common difference \(d = -\frac{3}{4}\).

Step 2: Define the Recursive Formula Structure

An arithmetic sequence can be represented by a recursive formula of the form: \(a_n = a_{n-1} + d\), where \(d\) is the common difference. We already found \(d = -\frac{3}{4}\).

Step 3: Write the Recursive Formula

Using the common difference identified in the previous steps and the recursive formula structure, the specific recursive formula for the given sequence is: \[ a_n = a_{n-1} - \frac{3}{4} \]. Additionally, since sequence elements need a starting point, we can specify that \(a_1 = -\frac{1}{2}\).

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This type of sequence is widespread in mathematics since it follows a linear pattern. It's defined by the fact that you can add the same value, known as the common difference, to each term to get the next term. A few key characteristics include:

• Linear nature of change from one term to another
• A straightforward formula for finding any term
• Dependence on the starting term and the common difference

To illustrate, consider the sequence \(-\frac{1}{2}, -\frac{5}{4}, -2, \ldots\). By identifying the common difference, this sequence can be expressed through a simple recursive formula.

Common Difference

The common difference in an arithmetic sequence is the fixed amount that you add (or subtract) to get from one term to the next. Determining this value is crucial for constructing the sequence's formula. Here’s how you identify it:

• Select any two consecutive terms in the sequence.
• Subtract the first term from the second term.

For instance, in the sequence \(-\frac{1}{2}, -\frac{5}{4}, -2, \ldots\), we observe the difference from \(-\frac{1}{2}\) to \(-\frac{5}{4}\). Calculating this gives us a common difference, \(d = -\frac{5}{4} - (-\frac{1}{2}) = -\frac{3}{4}\). This constant value is what characterizes the step size of the sequence.

Sequence Terms

In an arithmetic sequence, each number that appears in the progression is known as a term. Understanding the sequence terms is vital for determining the specific order and relationship between numbers in the sequence. Each term can be retrieved by applying the recursive formula:

• Use the recursive formula \(a_n = a_{n-1} + d\) to find the next term, where \(d\) is the common difference.
• Repeat this process to obtain as many terms as needed.

In our example, calculating further terms means starting with \(a_1 = -\frac{1}{2}\) and repeatedly adding \(-\frac{3}{4}\). This structured approach simplifies predicting any term in the sequence.

Starting Point in Sequences

Every arithmetic sequence begins with an initial term, which acts as a reference point for generating subsequent terms. This starting point is crucial as it sets the entire sequence's progression path. In mathematical terms:

• The first term is often denoted as \(a_1\).
• This initial term, combined with the common difference, constructs the entire sequence.

For our sequence \(-\frac{1}{2}, -\frac{5}{4}, -2, \ldots\), the starting point is \(a_1 = -\frac{1}{2}\). Knowing this allows us to apply the recursive formula to predict or backtrack in the sequence accurately. Starting from \(a_1\) and using the rule \(a_n = a_{n-1} - \frac{3}{4}\), each subsequent term can be systematically calculated.