Problem 3 - Algebra 2

Question

STANDARDIZED TEST PRACTICE What is the missing term in the geometric sequence: \(-\frac{1}{4}, \frac{1}{2},-1,2,-?\) \(\mathbf{A}-4\) \(\mathbf{B}-3 \frac{1}{2}\) \(\mathrm{C} 3 \frac{1}{2}\) \(\mathbf{D} 4\)

Step-by-Step Solution

Answer

The missing term is \(-4\). Option A is correct.

Step 1: Identify the common ratio

In a geometric sequence, each term is the product of the previous term and a constant ratio, called the common ratio (r). To find the common ratio, divide the second term by the first term: \( r = \frac{1}{2} \div \left(-\frac{1}{4}\right) = -2 \). The common ratio is \(-2\).

Step 2: Verify the sequence

Verify that the common ratio \(-2\) applies by multiplying each term by \(-2\) to get the next term: \( -\frac{1}{4} \times -2 = \frac{1}{2} \), \( \frac{1}{2} \times -2 = -1 \), \( -1 \times -2 = 2 \). The pattern holds for all the given terms so far.

Step 3: Find the missing term

To find the missing term, multiply the last known term by the common ratio. The last known term is \(2\), and the common ratio is \(-2\). So, \(2 \times -2 = -4\). Therefore, the missing term is \(-4\).

Step 4: Match with provided options

The sequence shows the missing term as \(-4\), which matches option \(\mathbf{A}\). Therefore, option \(\mathbf{A}\) is the correct answer.

Key Concepts

Common Ratio

In a geometric sequence, one of the key concepts you need to understand is the common ratio. This is the constant factor you multiply by each term to get the next term in the sequence. It's important to remember that this ratio remains the same throughout the sequence, allowing for a consistent pattern.
The common ratio is calculated by dividing any term by its preceding term. For instance, if you have terms like

• Term 1: \(-\frac{1}{4}\)
• Term 2: \(\frac{1}{2}\)

You find the common ratio ( ) by performing the division: \( = \frac{1}{2} \div \left(-\frac{1}{4}\right) = -2\).
Knowing the common ratio helps predict subsequent terms and solve for unknown values in the sequence.

Algebra

Algebra plays a crucial role in working with geometric sequences. It's the language we use to express general formulas, such as finding the common ratio or calculating terms. Understanding the basic algebraic manipulation allows you to solve these kinds of problems with ease.
For example, when you're asked to find the common ratio or a missing term, you convert words into mathematical expressions. In the case of our sequence, we converted the instruction "divide the second term by the first term" into the expression \(\frac{1}{2} \div \left(-\frac{1}{4}\right)\).
Remember that algebra isn't just about solving equations. It's also about understanding relationships between numbers, like those found in geometric patterns. This comprehension helps in identifying sequences and in verifying the correctness of your solutions.

Term Calculation

Calculating terms in a geometric sequence involves using the common ratio effectively. Once you know the common ratio, you can easily extend the sequence or find missing terms.
To calculate the missing term in a sequence, you take the last known term and multiply it by the common ratio. Continuing with our example, the last known term is 2, and the common ratio is \-2\. Therefore, the missing term is calculated as follows: \(2 \times -2 = -4\).
This calculation tells you that the missing term is \-4\. Such calculations are vital in geometric sequences because they allow for predictions and validations of sequences. Using this method, you can fill in gaps regardless of how many terms in the sequence are missing.