Problem 24 - College Algebra

Question

For the following exercises, write a recursive formula for each geometric sequence. $$ a_{n}=\\{-1,5,-25,125, \ldots\\} $$

Step-by-Step Solution

Answer

The recursive formula is \(a_1 = -1\), \(a_n = a_{n-1} \times (-5)\).

Step 1: Identifying the First Term

To write a recursive formula for a geometric sequence, we first need to identify the first term in the sequence. Looking at the given terms \(a_1 = -1\), so the first term \(a_1 = -1\).

Step 2: Determining the Common Ratio

Next, we need to find the common ratio \(r\) for the sequence. The common ratio is the number that each term is multiplied by to get the next term. Take the second term divided by the first term \(r = \frac{5}{-1} = -5\). We can verify this ratio holds for subsequent terms, such as \(-25 \div 5 = -5\) and \(125 \div -25 = -5\). Hence, the common ratio \(r = -5\).

Step 3: Writing the Recursive Formula

With the first term \(a_1 = -1\) and the common ratio \(r = -5\), we can write the recursive formula for the sequence as follows: \(a_1 = -1\), and for \(n \geq 2\), \(a_n = a_{n-1} \times (-5)\).

Key Concepts

Geometric Sequence

A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This type of sequence frequently appears in mathematics and can be easily identified by this repetitive multiplication process.

• The sequence begins with an initial term, known as the first term.
• Every subsequent term is the product of the previous term and the common ratio.

For example, consider the sequence \(-1, 5, -25, 125, \ldots \). Here, each term is multiplied by \(-5\) to form the next, establishing this series as a geometric sequence. Understanding this foundational pattern is crucial for working with recursive formulas.

Common Ratio

The common ratio in a geometric sequence is the constant factor between consecutive terms. It determines how the sequence evolves from one term to the next. Identifying the common ratio is vital for constructing recursive formulas accurately.

• Calculate it by dividing any term by the previous term.
• In our example, dividing the second term \(5\) by the first term \(-1\) yields \(-5\).
• Verification can be done by repeating this division with other terms, such as \(-25 \div 5\) and \(125 \div -25\), both resulting in \(-5\).

Once confirmed, mentioning the common ratio in the recursive formula helps in systematically generating subsequent terms of the sequence.

Recursive Sequence

A recursive sequence is one where each term is defined based on the previous term(s). In the context of geometric sequences, recursion involves using the common ratio to obtain each subsequent term from its predecessor.

• One must know both the first term and the common ratio to construct the recursive formula.
• For \(n \geq 2\), each term \(a_n\) is expressed as \(a_{n-1} \times r\), where \(r\) is the common ratio.
• In our example, \(a_n = a_{n-1} \times (-5)\) for all \(n \geq 2\).

This approach underscores how interconnected terms are in a recursive sequence, making it a powerful tool for generating sequence elements.

First Term Identification

Identifying the first term is the initial and crucial step in writing a recursive formula. This term serves as the starting point from which all other terms are derived in a sequence.

• To identify it, look at the sequence’s notation \(a_1\).
• In the example \(-1, 5, -25, 125, \ldots\), \(a_1 = -1\).
• The first term not only initiates the sequence but also anchors the recursive formula, allowing all subsequent terms to be calculated accurately.

Proper identification is essential for establishing the framework of the sequence, ensuring that each step follows the correct mathematical path dictated by the sequence's parameters.