Problem 24 - College Algebra with Corequisite Support

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 \(a_n = a_{n-1} \cdot (-5)\).

Step 1: Identify the First Term

The first term in a geometric sequence is denoted by \(a_1\). From the given sequence \(\{-1, 5, -25, 125, \ldots\}\), the first term \(a_1 = -1\).

Step 2: Determine the Common Ratio

The common ratio in a geometric sequence can be found by dividing the second term by the first term. Thus, \(r = \frac{5}{-1} = -5\). Therefore, the sequence has a common ratio of \(-5\).

Step 3: Write the Recursive Formula

A recursive formula for a geometric sequence is generally expressed as \(a_n = a_{n-1} \cdot r\), where \(a_1\) is the first term. Substituting the known values into this formula:\[ a_n = a_{n-1} \cdot (-5) \] This formula allows you to find any term \(a_n\) in the sequence using the previous term \(a_{n-1}\).

Key Concepts

recursive formula

A recursive formula in the context of geometric sequences is a way of defining terms in terms of previous ones. It provides a method to generate each term from the one before it, rather than independently calculating each term from scratch. This approach follows a natural sequence that builds progressively as you proceed through the list of terms.

In simple terms, if you know a specific term of the sequence, you can find the next one by applying the recursive formula. The general recursive formula for a geometric sequence looks like this:

• Start with the first term, denoted by \( a_1 \).
• Use the formula \( a_n = a_{n-1} \cdot r \) to find subsequent terms.

In this formula, \( a_n \) represents the term you want to find, \( a_{n-1} \) is the term before it, and \( r \) is the common ratio. Using the recursive formula helps in exploring sequences without recalculating each term from the beginning.

common ratio

The common ratio is an essential element in geometric sequences, as it acts as the multiplier between consecutive terms. This ratio is constant throughout the sequence, which guarantees the geometric nature of the series. To find the common ratio, you take any term in the sequence and divide it by its preceding term.

In the sequence \(\{-1, 5, -25, 125, \ldots\} \) provided in the exercise, to find the common ratio:

• Identify the second term and the first term.
• Divide them: \( r = \frac{5}{-1} \), which results in \(-5\).

This indicates that each term is obtained by multiplying the previous one by \(-5\). The uniformity of this ratio across all terms ensures that the sequence maintains its geometric properties.

first term

In any geometric sequence, the first term is where it all begins. It is the initial value from which every other term is derived by applying the common ratio iteratively. Knowing the first term is crucial for determining the entire sequence, especially when using recursive formulas.

For the sequence \(\{-1, 5, -25, 125, \ldots\} \) in the exercise, the first term \( a_1 \) is \(-1\). Using this first term:

• You anchor the sequence's start.
• Apply the common ratio \( r \) to find the next terms.

Thus, the first term serves as the reference point for generating all subsequent terms in the sequence, establishing the foundational step needed to use the recursive formula properly.