Problem 11 - Algebra 2 and Trigonometry

Question

In \(3-14,\) determine whether each given sequence is geometric. If it is geometric, find \(r\) . If it is not geometric, explain why it is not. $$ 1,-10,100,-1,000,10,000, \ldots $$

Step-by-Step Solution

Answer

The sequence is geometric with a common ratio \(r = -10\).

Step 1: Identify Terms of the Sequence

The given sequence is \(1, -10, 100, -1000, 10000, \ldots\). These terms appear to follow a pattern in their values.

Step 2: Check for a Common Ratio

In a geometric sequence, each term is obtained by multiplying the previous term by a constant. Check if there is a common ratio \(r\) by dividing each term by its preceding term: \[\frac{-10}{1} = -10, \quad \frac{100}{-10} = -10, \quad \frac{-1000}{100} = -10\] It appears that each successive term is obtained by multiplying the previous term by \(-10\).

Step 3: Confirm the Geometric Nature

Since the same ratio \(-10\) is consistent between each pair of consecutive terms, the sequence is indeed geometric. This common ratio is a defining characteristic of geometric sequences.

Step 4: Declare and Explain the Common Ratio

Since the common ratio is the same throughout the sequence, we can conclude that this sequence is geometric. The common ratio \(r\) is \(-10\).

Key Concepts

Understanding the Common Ratio in Geometric Sequences

In a geometric sequence, the core element that distinguishes it is the presence of a constant that links each term to the next. This constant is known as the **common ratio**. Essentially, if each term is derived by multiplying the previous one with a uniform number, then this number is what we refer to as the common ratio.

Here's how you can understand the formula better:

• Take the second term and divide it by the first term.
• Repeat this process for subsequent terms to confirm consistency.
• The result should be the same for each consecutive pair of terms.

For example, in the sequence provided: \(1, -10, 100, -1000, 10000, \ldots\), dividing \(-10\) by \(1\) gives \(-10\), and similarly, dividing \(100\) by \(-10\) also gives \(-10\), continuing this pattern confirms the consistency of the common ratio. So, here our common ratio \(r\) is \(-10\).

Knowing the common ratio not only helps in identifying the sequence type but also aids in predicting future terms with ease.

Exploring Sequence Patterns

A sequence is a succinct way of organizing numbers in a particular order according to a specific rule or pattern. Recognizing these patterns is crucial for understanding both simple and complex sequences. Geometric sequences showcase one such pattern, where each term is related to its predecessor by a multiplication factor, known as the common ratio.

• **Definition**: A geometric sequence continues by multiplying a fixed, non-zero number, known as the common ratio, by each term to produce the next.
• **Recognition**: Examine the series of numbers—if a consistent multiplication operation defines the progression from one term to the next, that's a geometric sequence.
• **Example Application**: Recognizing the sequence pattern in our example with numbers \(1, -10, 100, -1000, 10000, \ldots\) becomes a straightforward task. You spot the multiplication pattern of \(-10\) fairly quickly after examining a few terms.

This kind of sequential pattern recognition is pivotal in enhancing a student's problem-solving capabilities, especially when extending sequences or constructing formulas.

Geometric Sequences in Algebra 2

In Algebra 2, geometric sequences form a fundamental part of understanding series and patterns. They provide a structured way of approaching problems where each term is systematically derived from the previous ones through multiplication by a fixed ratio.

Some key points to keep in mind while dealing with geometric sequences in Algebra 2 include:

• **Formulating**: Often, you'll be asked to identify whether a sequence is geometric. Use the common ratio test to make this determination.
• **Properties**: All terms, starting from the second one, are constructed by multiplying their predecessor with the same common ratio.
• **Significance**: Recognizing geometric sequences opens the door to solving real-world problems, particularly those involving exponential growth and decay scenarios.
• **General Formula**: The \(n\)-th term of a geometric sequence can be found using the formula \(a_n = a_1 \cdot r^{(n-1)}\), where \(a_1\) is the first term, and \(r\) is the common ratio.

In our example, applying the concept of geometric sequences, we established that each term is derived by consistently multiplying by \(-10\), revealing the intrinsic structure beneath the series of numbers.