Problem 11 - Algebra and Trigonometry Real Mathematics, Real People

Question

Identifying a Geometric Sequence Determine whether or not the sequence is geometric. If it is, find the common ratio.Identifying a Geometric Sequence Determine whether or not the sequence is geometric. If it is, find the common ratio. $$1,-\frac{1}{2}, \frac{1}{4},-\frac{1}{8}, \dots$$

Step-by-Step Solution

Answer

The given sequence is a geometric sequence, with a common ratio of -1/2.

Step 1: Identify the pattern

An examination of the given sequence reveals a pattern of alternating positive and negative numbers. Furthermore, each new term seems to be the previous term divided by -2. This suggests it might be a geometric sequence.

Step 2: Testing for common ratio

If indeed this is a geometric sequence, then any term divided by its previous term should equal the same number. Let's test this hypothesis: \n \(-\frac{1}{2} \div 1 = -\frac{1}{2}, \frac{1}{4} \div -\frac{1}{2} = -\frac{1}{2}, -\frac{1}{8} \div \frac{1}{4} = -\frac{1}{2}, ... \) \n The result shows that every term divided by the previous one equals -1/2, which confirms that the sequence is geometric.

Step 3: Determine the common ratio

The ratio, or the number each term is multiplied by to get the next term, is -1/2. This can be deduced from the tests carried out in the previous step.

Key Concepts

Common Ratio

Understanding the concept of a common ratio is essential when dealing with geometric sequences. A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In simple terms, if you start with a number and consistently multiply by the same factor to get the next number in the series, you are dealing with a geometric sequence. The common ratio is the key to recognizing this type of sequence.

For example, let's consider the geometric sequence provided in the exercise: \(1, -\frac{1}{2}, \frac{1}{4}, -\frac{1}{8}, \dots\). To find the common ratio, we divide any term by its preceding term (after the first). Doing this consistently, as shown in the original problem's solution, we get \(-\frac{1}{2}\) each time, confirming that the sequence does have a common ratio and is therefore geometric.

Pattern Recognition in Sequences

Pattern recognition plays a crucial role in identifying geometric sequences and understanding their properties. When initially examining a sequence, it's important to observe whether there is a repeated operation from one term to the next. This could be a consistent addition, subtraction, multiplication, or division. For geometric sequences in particular, you're looking for a constant factor between consecutive terms.

Let's examine the provided sequence: \(1, -\frac{1}{2}, \frac{1}{4}, -\frac{1}{8}, \dots\). The observable pattern here is not only the change of signs between consecutive terms but also the halving of each term's absolute value. This suggests that the sequence could be geometric with a negative common ratio since each term results from multiplying the previous one by \(-\frac{1}{2}\). Recognizing such patterns is vital, as it informs us about the nature of the sequence and how it progresses. This predictive capability is at the heart of sequence analysis.

Arithmetic Operations with Fractions

Arithmetic operations with fractions are often encountered when working with geometric sequences, especially when determining the common ratio. It's crucial to be comfortable with these operations, as they are used to divide one term by another to find the ratio.

For instance, in the given sequence, to verify if it is geometric, we perform division of terms. Here's a breakdown: taking the second term \(-\frac{1}{2}\) and dividing it by the first term \(1\), we get \(-\frac{1}{2}\). Similarly, dividing the third term \(\frac{1}{4}\) by the second \(-\frac{1}{2}\) yields the same ratio. The operation is straightforward: when dividing fractions, you multiply by the reciprocal of the divisor. Recognizing this simple operation allows us to handle even more complex fraction-based sequences, establishing clarity and confidence in identifying the common ratio.