Problem 33 - College Algebra with Corequisite Support

Question

For the following exercises, write a recursive formula for each arithmetic sequence. $$ a=\\{8.9,10.3,11.7, \ldots\\} $$

Step-by-Step Solution

Answer

The recursive formula is \( a_n = a_{n-1} + 1.4 \) with \( a_1 = 8.9 \).

Step 1: Identify the Terms

The given sequence is \( a = \{8.9, 10.3, 11.7, \ldots\} \). The first term \( a_1 = 8.9 \). We need to find the common difference \( d \) to create the recursive formula.

Step 2: Calculate the Common Difference

To find the common difference \( d \), subtract the first term from the second term: \( d = 10.3 - 8.9 = 1.4 \). Thus, each term in the sequence increases by 1.4.

Step 3: Write the Recursive Formula

In a recursive formula for an arithmetic sequence, each term is defined in terms of the previous term. Since we know the first term \( a_1 = 8.9 \) and the common difference \( d = 1.4 \), the recursive formula is:\[ a_n = a_{n-1} + 1.4 \text{ for } n \geq 2, \text{ with } a_1 = 8.9 \]

Key Concepts

Recursive Formula

In arithmetic sequences, a recursive formula is a way to define the sequence's terms by expressing each term as a function of the previous term. It's like a rule you can follow to get from one number to the next in the sequence. Instead of listing all the numbers in a sequence, you just need to know how to compute any term using its predecessor.
This type of formula is particularly useful for sequences where listing out all terms is impractical. In a recursive formula, you usually specify the first term separately, as it acts as the starting point. Then, you use a mathematical rule involving the previous term to find each successive term.
For example, in the sequence given:

• The first term is noted as \( a_1 = 8.9 \).
• To find any term \( a_n \), use the formula \( a_n = a_{n-1} + 1.4 \), where \( n \geq 2 \).

This means you add 1.4 to the previous term to get the next term in the series.

Common Difference

The common difference in an arithmetic sequence is the value that separates one term in the sequence from the next. It's a constant that you add to each term to get the following term. This is what defines the uniformity of the sequence, making it predictable and easier to work with.
To find the common difference, you can simply subtract any term from the following term. For instance, with the sequence provided:

• Subtract the first term (\(8.9\)) from the second term (\(10.3\)) to get \( d = 10.3 - 8.9 = 1.4 \).

This calculation shows that the sequence increases by \(1.4\) each time. Having the common difference is crucial for writing both recursive and explicit formulas for the sequence.

Sequence Terms

Sequence terms are the individual elements or numbers that form the complete arithmetic sequence. Each term in an arithmetic sequence is related to its position in the sequence and can be calculated using a formula.
The most important thing about terms in a sequence is their relationship to one another, typically defined by the common difference. Understanding this relationship helps you determine any term in the sequence efficiently without the need to list them all.

• The first term in the sequence, often denoted as \( a_1 \), serves as the foundation.
• Each subsequent term \( a_n \) can be defined using the previous term and the common difference, using the recursive formula.

For example, in the sequence \( a = \{8.9, 10.3, 11.7, \ldots \} \), each term increases from the previous one by \( 1.4 \). Knowing the foundation (first term) and the rule (common difference) allows you to generate the sequence indefinitely.