Problem 9 - 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. $$ 1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \dots $$

Step-by-Step Solution

Answer

The sequence is geometric with a common ratio \( r = \frac{1}{3} \).

Step 1: Understanding a Geometric Sequence

A sequence is geometric if there is a constant ratio between consecutive terms. This means for every term in the sequence, the next term can be found by multiplying the previous term by the same constant ratio, denoted as \( r \).

Step 2: Calculate the Ratio for Consecutive Terms

We find the ratio \( r \) by dividing the second term by the first term: \( r = \frac{1/3}{1} = \frac{1}{3} \). To confirm, calculate the ratio for the next pair of terms: \( r = \frac{1/9}{1/3} = \frac{1}{3} \). This is consistent so far.

Step 3: Continue Checking the Ratio

Check the ratio between the next consecutive terms to ensure consistency: \( r = \frac{1/27}{1/9} = \frac{1}{3} \). The ratio remains the same.

Step 4: Determine If the Sequence is Geometric

Since the ratio \( r = \frac{1}{3} \) is constant between all consecutive terms in the given sequence, this sequence is geometric.

Key Concepts

Understanding the Common Ratio in Geometric Sequences

A geometric sequence is a fascinating mathematical concept where each term is derived by multiplying the previous one with a fixed number called the common ratio.
This gives these sequences a uniformity that linear sequences do not possess. The common ratio, denoted as \( r \), is fundamental in forming and analyzing geometric sequences. To find this ratio, you divide any term in the sequence by the term preceding it. For example, in the sequence given:

• The first term is \( 1 \), and the second term is \( \frac{1}{3} \).
• The common ratio can be calculated as \( r = \frac{1/3}{1} = \frac{1}{3} \).

A consistent common ratio confirms the sequence's geometric property, allowing us to predict future terms easily. This predictability makes geometric sequences much easier to work with, especially in algebra and calculus.

Sequence Analysis: Is the Sequence Geometric?

Analyzing a sequence involves checking if it holds the properties of a specific type, such as arithmetic, geometric, or neither. In the case of geometric sequences, we look for a constant ratio:
The given sequence \( 1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \dots \) requires examining the ratio between each pair of terms:

• From the first term to the second: \( \frac{1}{3} \)
• From the second term to the third: \( \frac{1}{9} \div \frac{1}{3} = \frac{1}{3} \)
• From the third term to the fourth: \( \frac{1}{27} \div \frac{1}{9} = \frac{1}{3} \)

The persistence of \( \frac{1}{3} \) as the ratio shows that this sequence is indeed geometric. Such consistency is crucial, as any deviation would imply the sequence is not geometric. Understanding whether a sequence maintains a common ratio allows for accurate predictions of unseen terms and is foundational in mathematical reasoning.

Using Mathematical Reasoning to Verify Geometric Sequences

Mathematical reasoning involves using logic and structured thought processes to solve problems. When applied to sequence verification, it entails the step-by-step examination of the sequence's properties.
For the sequence in question, initially, you determine whether it displays a geometric nature by calculating a common ratio between each term pair.

• The ratio between any consecutive terms must consistently yield the same value. In our sequence, \( \frac{1}{3} \) consistently appears, affirming its geometric nature.

Next, we conclude with certainty about the sequence's nature because the common ratio correctly identifies it as geometric.

• This logical deduction aids not only in verifying sequences but in predicting or expanding them.

Mathematical reasoning ensures accuracy in mathematical practices and a deeper understanding of how sequences relate within algebraic contexts. It is the foundation upon which much of mathematical analysis is built, underscoring the importance of recognizing patterns like geometric sequences.