Problem 8 - 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. $$ 36,12,4, \frac{4}{3}, \dots $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Concept of a Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio \(r\). To determine if a sequence is geometric, the ratio between consecutive terms should be the same throughout the sequence.

Step 2: Calculate the Ratio Between the First and Second Terms

Identify the first two terms of the sequence: 36 and 12. Calculate the ratio \(r\) as follows:\[ r = \frac{12}{36} = \frac{1}{3} \]

Step 3: Calculate the Ratio Between the Second and Third Terms

Identify the second and third terms of the sequence: 12 and 4. Calculate the ratio \(r\) as follows:\[ r = \frac{4}{12} = \frac{1}{3} \]

Step 4: Calculate the Ratio Between the Third and Fourth Terms

Identify the third and fourth terms of the sequence: 4 and \(\frac{4}{3}\). Calculate the ratio \(r\) as follows:\[ r = \frac{\frac{4}{3}}{4} = \frac{1}{3} \]

Step 5: Conclusion: Consistent Ratio

Since the ratio \(r\) is consistent (\(\frac{1}{3}\)) between each pair of consecutive terms, the sequence 36, 12, 4, \(\frac{4}{3}, \dots\) is indeed a geometric sequence.

Key Concepts

Understanding Common Ratio in a Geometric Sequence

One of the defining characteristics of a geometric sequence is its common ratio. This is the fixed number, denoted by \( r \), by which we multiply each term to get the next.
The equation to find the common ratio \( r \) between two consecutive terms \( a_{n} \) and \( a_{n+1} \) is given by:

• \( r = \frac{a_{n+1}}{a_{n}} \)

In our example, the sequence \( 36, 12, 4, \frac{4}{3}, \dots \) has been checked. As demonstrated, the ratio between each pair of consecutive terms is \( \frac{1}{3} \).
This consistent value of \( r = \frac{1}{3} \) confirms it's a geometric sequence. This consistency of the ratio is essential—the common ratio must stay the same throughout the sequence for it to remain geometric.

The Nature of Sequences

A sequence is an ordered list of numbers. In mathematics, understanding different types of sequences is key to mastering more complex concepts. There are several types of sequences:

• Arithmetic sequences: where each term is obtained by adding a constant.
• Geometric sequences: where each term is obtained by multiplying by a constant known as the common ratio.
• Other sequences may follow specific patterns or rules.

Geometric sequences are particularly interesting because the terms grow or shrink at a specific rate determined by that common ratio \( r \).
This exponential nature can dictate the behavior of functions and models using such sequences, making them crucial in areas like finance, physics, and various other fields.

Geometric Sequences in Algebra 2

In Algebra 2, you delve deeper into sequences, utilizing them to form and solve equations.
Understanding geometric sequences involves:

• Identifying the first term and the common ratio \( r \).
• Applying the formula for the nth term \( a_n = a_1 \cdot r^{n-1} \), where \( a_1 \) is the first term and \( n \) is the term number.
• Using sequences to solve related algebraic problems.

Finding the common ratio, as in our example, allows you to determine future terms, sum parts of the sequence, or even solve for unknowns in applications.
These skills are fundamental in Algebra 2 and are applied across various mathematical problems, helping students grasp more nuanced concepts in math.