Problem 23 - Algebra for College Students

Question

Find the common ratio of the geometric sequence with 3 rd term 12 and 6 th term \(96.2\)

Step-by-Step Solution

Answer

The common ratio is approximately 2.

Step 1: Understand the Geometric Sequence Formula

In a geometric sequence, each term is obtained by multiplying the previous term by a constant value called the common ratio, denoted as \( r \). The general formula for the \( n \)-th term of a geometric sequence is \( a_n = a_1 \cdot r^{n-1} \), where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.

Step 2: Write Equations Using Given Terms

We are given that the 3rd term \( a_3 = 12 \) and the 6th term \( a_6 = 96.2 \). Using the formula for the n-th term:\[ a_3 = a_1 \cdot r^2 = 12 \]\[ a_6 = a_1 \cdot r^5 = 96.2 \]

Step 3: Express Terms in Terms of Common Ratio

From the two equations:\[ a_1 \cdot r^2 = 12 \] \[ a_1 \cdot r^5 = 96.2 \]We can express \( a_1 \) in terms of the common ratio from one of these equations.

Step 4: Divide the Equations to Find \( r \)

To eliminate \( a_1 \) and find \( r \), divide the second equation by the first:\[ \frac{a_1 \cdot r^5}{a_1 \cdot r^2} = \frac{96.2}{12} \]This simplifies to:\[ r^{5-2} = \frac{96.2}{12} \]\[ r^3 = 8.0167 \]

Step 5: Solve for the Common Ratio \( r \)

To solve for \( r \), take the cube root of both sides of the equation:\[ r = \sqrt[3]{8.0167} \]Using a calculator, we find:\[ r \approx 2 \]

Key Concepts

Common Ratio in Geometric Sequences

In a geometric sequence, the common ratio is the factor by which we multiply to get from one term to the next. This ratio is a cornerstone of geometric sequences. It determines how the sequence progresses, whether it grows, decays, or stays constant. For example, if the common ratio is greater than 1, each term grows larger than the previous one. Conversely, if it's between 0 and 1, each term becomes smaller.

• In the exercise provided, we were tasked with finding the common ratio of a sequence where the 3rd term is 12 and the 6th is 96.2. Identifying the common ratio helps us understand the relationship between these terms clearly.

Understanding the Geometric Sequence Formula

The geometric sequence formula is like a map for navigating sequences. This formula, given by \( a_n = a_1 \cdot r^{n-1} \), allows us to find any term in the sequence.

• \( a_n \) is the term we are looking for.
• \( a_1 \) is the first term of the sequence.
• \( r \) is the common ratio.
• \( n \) is the position of the term we want.

A key insight is how each term is built upon its predecessor, amplified by the common ratio. By knowing this, you can create, dissect, or predict sequences effectively! Understanding this relationship helps in solving problems efficiently and accurately.

Finding a Term in a Sequence

Calculating a specific term in a sequence might seem tough at first, but with a grasp of the basics, it becomes straightforward.In the scenario with the 3rd term as 12 and 6th as 96.2, we utilize the geometric sequence formula to express given terms. \[ a_3 = a_1 \cdot r^2 = 12 \] \[ a_6 = a_1 \cdot r^5 = 96.2 \]

• This approach breaks down the problem; it makes it easier to see how each term connects through the common ratio.

From this, solving for the common ratio can lead to finding any other unknown terms in the sequence by just plugging values back into the formula.

A Step-by-Step Approach

Breaking down problems into steps is an efficient strategy in mathematics. In the given exercise, finding the common ratio was a step-by-step process. Here’s a quick overview:

• Step 1: Write equations using known terms.
• Step 2: Divide these equations to simplify and find the common ratio.
• Step 3: Solve for the common ratio by simplifying further, using \( r^3 \).

Using these steps eliminates variables systematically, leading us directly to our goal. The practice of breaking tasks into smaller steps not only makes complex problems manageable but also strengthens understanding by focusing attention on each piece of the solution.