Problem 34 - College Algebra with Corequisite Support

Question

For the following exercises, write a recursive formula for each arithmetic sequence. $$ a=\\{-0.52,-1.02,-1.52, \ldots\\} $$

Step-by-Step Solution

Answer

Recursive formula: \(a_1 = -0.52\), \(a_n = a_{n-1} - 0.50\) for \(n \geq 2\).

Step 1: Identify the Common Difference

In an arithmetic sequence, the difference between consecutive terms is constant. To find the common difference \(d\), subtract the first term from the second term: \[-1.02 - (-0.52) = -1.02 + 0.52 = -0.50\] So, the common difference is \(d = -0.50\).

Step 2: Define the Terms of the Sequence

Let's define the terms of the sequence \(a_n\) where \(a_1 = -0.52\). Each term in the sequence can be calculated using the previous term and the common difference. The general form of the arithmetic sequence is \(a_n = a_{n-1} + d\).

Step 3: Write the Recursive Formula

Using the first term and the common difference, the recursive formula for the sequence is: \[a_1 = -0.52\] \[a_n = a_{n-1} - 0.50\] for \(n \geq 2\). This formula allows you to find any term in the sequence if you know the previous term.

Key Concepts

Recursive Formula

A recursive formula is a tool that helps describe a sequence by expressing each term as a function of its preceding terms. For arithmetic sequences, the recursive formula provides a way to easily compute each term, as long as the previous term and the common difference are known. This form of a formula establishes an ongoing relationship between terms in the sequence.

In an arithmetic sequence, the recursive formula can be represented as:

• The first term, denoted as: \[ a_1 \]
• Every subsequent term: \[ a_n = a_{n-1} + d \]for \( n \geq 2 \)

Here, \( a_n \) represents the term you are trying to find, \( a_{n-1} \) is the term before it, and \( d \) is the common difference. This simple relationship makes recursive formulas particularly helpful when you need to calculate multiple terms in the sequence without direct computation from the first term each time.

Common Difference

In the world of arithmetic sequences, the common difference \( d \) is a core concept that defines the change from one term to the next. It's a constant value representing how much you add or subtract to get each subsequent term.

To identify the common difference:

• Subtract the first term from the second term:
• For example, given the sequence \{-0.52, -1.02, -1.52, \ldots\}
• Calculate: \[ d = -1.02 - (-0.52) = -1.02 + 0.52 = -0.50 \]

Here, the common difference is \(-0.50\). It means that each term is 0.50 less than the previous term. This characteristic of arithmetic sequences ensures that the pattern remains consistent throughout. Knowing the common difference allows you to quickly generate terms of the sequence or verify the sequence's validity.

Sequence Terms

The sequence terms are the individual numbers that make up an arithmetic sequence. Each term holds a specific position in the sequence and follows the established pattern set by the common difference.

In arithmetic sequences:

• The first term is often provided directly, which serves as the basis for finding the rest of the terms.
• For example: \[ a_1 = -0.52 \]
• Subsequent terms can be calculated using the recursive formula: \[ a_n = a_{n-1} + d \]

Starting from this first term, each next term in our example sequence \{-0.52, -1.02, -1.52, ...\} is found by reducing the previous term by the common difference of \(-0.50\). This method of establishing sequence terms is efficient and easy, making it simple to determine any term’s position within the sequence.