Problem 65 - Algebra 2

Question

Find the missing term in the geometric sequence \(8, \square, 0.5,-0.125, \dots\) $$ \begin{array}{lllll}{\text { A. } 2} & {\text { B. }-2} & {\text { C. } 4} & {\text { D. }-4}\end{array} $$

Step-by-Step Solution

Answer

The missing term in the geometric sequence is -2 (Option B).

Step 1: Identifying the Sequence

Examine the sequence: \(8, \square, 0.5, -0.125, \dots\). It's clear that this is a decreasing sequence where every term is obtained by multiplying the previous term by a certain constant. Hence, it is a geometric sequence.

Step 2: Find the Common Ratio

The common ratio of a geometric sequence is obtained by dividing any term by its previous term. We can apply this principle to the third and fourth terms of the sequence. So, the common ratio, r, is \(r = \frac{-0.125}{0.5} = -0.25\)

Step 3: Find the Missing Term

Knowing that the common ratio is -0.25, we can use this information to find the missing term. The second term in our sequence would be the first term multiplied by the common ratio, so the missing term can be found as: \(8 * -0.25 = -2 \). Thus, the missing term is -2.

Key Concepts

Common Ratio

In a geometric sequence, the **common ratio** is the constant factor that each term is multiplied by to get the next term in the sequence. Identifying the common ratio is crucial when dealing with problems involving geometric sequences because it helps us to determine unknown terms. For example, a sequence might look like this: 2, 6, 18, 54, and so forth. To find the common ratio here, you divide the second term by the first term:

• 6 divided by 2 equals 3.
• Similarly, 18 divided by 6 equals 3.
• 54 divided by 18 also equals 3.

In the given problem, we identify the common ratio by dividing one term by its preceding term, from the known terms of the sequence. This ensures every term in the sequence is connected by the same ratio, validating it as a geometric sequence.

Term Multiplication

**Term multiplication** is a key concept in geometric sequences. Each term is calculated by multiplying the previous term by the common ratio. Consider the sequence given in the problem: 8, ?, 0.5, -0.125. Once we know the common ratio, which is -0.25, we can easily find any missing terms by multiplying the previous term by this ratio. For example, starting with the first term, 8, we multiply by the common ratio of -0.25:

• 8 multiplied by -0.25 equals -2.

This gives us the missing term. Similarly, this process can continue for the other terms in a sequence, making sure that each term correctly adheres to the ratio established by its predecessor.

Decreasing Sequence

A **decreasing sequence** is a sequence where each subsequent term is smaller than the previous one. In a geometric sequence, this decrease happens when the common ratio is a positive number less than 1 or a negative fraction, as shown in the problem where the common ratio is -0.25. When each term is multiplied by a negative number less than 1 in absolute value, the sequence decreases. For instance, 8 multiplied by -0.25 becomes -2, a smaller value, and so on with subsequent terms. This characteristic can be used strategically to recognize similar patterns across different sequence problems.

Sequence Problem Solving

When it comes to **sequence problem solving**, understanding the core properties of geometric sequences is essential. This involves not just finding missing terms, but also analyzing the sequence for patterns and connections. Here's how you can tackle these problems:

• First, identify if the sequence is geometric by checking if there is a common ratio.
• Calculate this ratio by dividing a term by its previous term when possible.
• Utilize the common ratio to find missing terms by multiplying the known values accordingly.
• Always check your answer by verifying that all terms maintain a consistent common ratio to ensure your solution is accurate.

With these strategies and a good grasp of geometric sequences and their properties, students can improve their problem-solving skills and tackle exercises with confidence.