Problem 8 - College Algebra with Corequisite Support

Question

For the following exercises, find the common ratio for the geometric sequence. \(-2,-\frac{1}{2},-\frac{1}{8},-\frac{1}{32},-\frac{1}{128}, \ldots\)

Step-by-Step Solution

Answer

The common ratio is \(\frac{1}{4}\).

Step 1: Understand the Problem

We need to find the common ratio of the given geometric sequence. The sequence given is \(-2, -\frac{1}{2}, -\frac{1}{8}, -\frac{1}{32}, -\frac{1}{128}, \ldots\). In a geometric sequence, the common ratio is the factor by which we multiply one term to get the next term.

Step 2: Identify Successive Terms

For a geometric sequence, the common ratio \(r\) is found by dividing any term by the previous term. Let's identify two successive terms in the sequence to calculate this.

Step 3: Calculate the Common Ratio Using the First Two Terms

We use the first and second terms of the sequence to find the common ratio. Let the first term \(a_1 = -2\) and the second term \(a_2 = -\frac{1}{2}\). The formula for the common ratio \(r\) is: \(r = \frac{a_2}{a_1}\).

Step 4: Perform the Calculation

Substitute the identified values into the formula: \(r = \frac{-\frac{1}{2}}{-2}\). Simplify this to find \(r\).

Step 5: Simplify the Expression

Calculating \(\frac{-\frac{1}{2}}{-2}\) gives \(\frac{1}{4}\), since the negatives cancel each other out.

Step 6: Verify the Common Ratio

To check correctness, multiply the first term by \(\frac{1}{4}\) to see if we get the second term. \(-2 \times \frac{1}{4} = -\frac{1}{2}\). This confirms the common ratio calculation is correct, as the sequence follows from this ratio.

Key Concepts

Common Ratio

In a geometric sequence, understanding the concept of the common ratio is crucial. The common ratio is the factor by which we multiply one term to reach the next term in a sequence. This idea is central to the nature of geometric sequences, where each term after the first is the product of the previous term and the common ratio.
For example, in the sequence

• \(-2, -\frac{1}{2}, -\frac{1}{8}, -\frac{1}{32}, -\frac{1}{128}, \ldots\)

the process to find the common ratio is straightforward. We pick two successive terms, say

• \(-2\)
• \(-\frac{1}{2}\)

and divide the second by the first. This gives: \[ r = \frac{-\frac{1}{2}}{-2} = \frac{1}{4} \].
This means that each term is \(\frac{1}{4}\) of the previous term, establishing a consistent pattern throughout the sequence.
Identifying and calculating the common ratio is crucial since it helps us predict further terms in the sequence, ensuring we can build or continue the pattern accurately.

Mathematical Sequence

A mathematical sequence is a list of numbers arranged in a particular order following a certain rule. In this context, geometric sequences are a special type of mathematical sequence where each term after the first is derived by multiplying the prior term by a constant factor, known as the common ratio.
Let's break down the specific sequence we're examining:

• \(-2, -\frac{1}{2}, -\frac{1}{8}, -\frac{1}{32}, -\frac{1}{128}, \ldots\)

This is a geometric sequence because each term is a consistent fraction of the previous term. We have identified the common ratio as \(\frac{1}{4}\), indicating a systematic reduction.

Sequence Growth and Reduction

A key feature of this geometric sequence is reduction, as each term becomes increasingly smaller and approaches zero. Such sequences illustrate how rapid decrement happens over progression.

• The choices of terms are infinite in such sequences.
• Each term provides a reliable basis for predicting the next.

Understanding these sequences enables mathematical curiosity and appreciation of patterns within mathematics.

Algebraic Calculation

Algebraic calculations play a pivotal role in understanding geometric sequences. By utilizing algebra, we can not only calculate terms but derive equations capturing the sequence's behavior.
In the given exercise, we employed algebraic methods to determine the common ratio \(r\), using the formula:

• \( r = \frac{a_2}{a_1} \)

where \(a_2\) and \(a_1\) are successive terms of the sequence. Crucial steps in calculating involved algebraic simplifications such as dividing fractions, i.e.,

• \( \frac{-\frac{1}{2}}{-2} \) simplifies to \( \frac{1}{4} \).

Part of mastering algebra in sequences includes ensuring each operation follows correct arithmetic rules. Simplifying accurately ensures consistent results.

Practical Checkpoint

Verifying the calculation was a crucial checkpoint. By multiplying

• \(-2 \) by \(\frac{1}{4}\)

we see it equates to \(-\frac{1}{2}\), proving our algebraic process was correct.
Mastery in algebraic calculations allows for not just solving specific sequences, but forming the basis for deeper exploration into more complex mathematical concepts.