Problem 25 - College Algebra

Question

For the following exercises, write a recursive formula for each geometric sequence. $$ a_{n}=\\{-32,-16,-8,-4, \ldots\\} $$

Step-by-Step Solution

Answer

The recursive formula is \( a_n = a_{n-1} \times \frac{1}{2} \) with \( a_1 = -32 \).

Step 1: Identify the First Term

The first term of the geometric sequence is the first number in the series. For this sequence, the first term \( a_1 = -32 \).

Step 2: Determine the Common Ratio

To find the common ratio \( r \), divide the second term by the first term. For this sequence, \( r = \frac{-16}{-32} = \frac{1}{2} \). Verify by dividing subsequent terms: \( \frac{-8}{-16} = \frac{1}{2} \) and \( \frac{-4}{-8} = \frac{1}{2} \).

Step 3: Write the Recursive Formula

The recursive formula for a geometric sequence is given by \( a_n = a_{n-1} \times r \). Substitute the first term and the common ratio into the formula: \( a_1 = -32 \) and \( r = \frac{1}{2} \). The recursive formula is: \( a_n = a_{n-1} \times \frac{1}{2} \), with \( a_1 = -32 \).

Key Concepts

Recursive Formula

A recursive formula is a way to define a sequence where each term is formulated in relation to the preceding one. In geometric sequences, each term is calculated by multiplying the previous term by a fixed number, known as the common ratio. This formula provides a quick and efficient way to find any term in the sequence without calculating all preceding terms individually.
For the given sequence, the recursive formula is expressed as:

• \( a_n = a_{n-1} imes r \)
• "\( a_n \)" represents the nth term in the sequence.
• "\( r \)" is the common ratio.

To utilize this formula effectively, you must first know the first term and the common ratio. Understanding the recursive formula is crucial because it allows you to predict how the sequence will evolve step by step.

Common Ratio

The common ratio \( r \) in a geometric sequence is critical as it determines the pattern of the sequence. It is the constant factor that we multiply by each term to get the next term. To find \( r \), divide any term by its preceding term.
For the sequence \(-32, -16, -8, -4, \ldots\), we can calculate:

• \( r = \frac{-16}{-32} = \frac{1}{2} \)

Verification is key. Consistency across the sequence should be checked:

• \( \frac{-8}{-16} = \frac{1}{2} \)
• \( \frac{-4}{-8} = \frac{1}{2} \)

This confirms \( r = \frac{1}{2} \) is correct for the sequence. The common ratio shapes the trajectory of the sequence allowing us to predict future terms.

First Term

Identify the first term, denoted as \( a_1 \), as it is the starting point of a geometric sequence. This initial term is essential because the recursive formula begins with it. It sets the stage for subsequent terms to be computed.
In our case:

• The first term \( a_1 = -32 \).

Knowing \( a_1 \) is fundamental when writing the recursive formula since it not only initiates the pattern but also validates each subsequent term calculated from the formula.
So, the confidence in determining future terms is directly tied to our understanding and identification of the first term of the sequence.