Problem 99 - A Graphical Approach to Precalculus with Limits

Question

CONCEPT CHECK Suppose that \(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, .\) is a geometric sequence. Is the sequence \(a_{1}, a_{3}, a_{5}, \dots\) geometric?

Step-by-Step Solution

Answer

Yes, the sequence \(a_1, a_3, a_5, \ldots\) is geometric with common ratio \(r^2\).

Step 1: Define a Geometric Sequence

A sequence is geometric if there is a common ratio \( r \) such that each term after the first is the product of the previous term and \( r \). Mathematically, for a sequence \( a_1, a_2, a_3, \ldots \), the terms satisfy \( a_{n+1} = a_n \cdot r \).

Step 2: Express General Terms of the Original Sequence

For the given sequence \( a_1, a_2, a_3, a_4, a_5, \ldots \), with common ratio \( r \), each term can be expressed as follows: \( a_1 = a_1 \), \( a_2 = a_1 \cdot r \), \( a_3 = a_1 \cdot r^2 \), \( a_4 = a_1 \cdot r^3 \), \( a_5 = a_1 \cdot r^4 \), and so on.

Step 3: Check if Subsequence Is Geometric

Consider the subsequence formed by the odd-indexed terms: \( a_1, a_3, a_5, \ldots \). These correspond to the terms \( a_1 = a_1 \), \( a_3 = a_1 \cdot r^2 \), \( a_5 = a_1 \cdot r^4 \), etc. Each term in this subsequence can be expressed using the even powers of \( r \).

Step 4: Determine the Common Ratio for the Subsequence

To find the common ratio for the new subsequence \( a_1, a_3, a_5, \ldots \), compute \( \frac{a_3}{a_1} = \frac{a_1 \cdot r^2}{a_1} = r^2 \) and \( \frac{a_5}{a_3} = \frac{a_1 \cdot r^4}{a_1 \cdot r^2} = r^2 \). This shows all adjacent terms in this subsequence have a common ratio \( r^2 \).

Step 5: Conclude Whether the Subsequence is Geometric

Since all adjacent terms in the sequence \( a_1, a_3, a_5, \ldots \) share the common ratio \( r^2 \), this subsequence is a geometric sequence.

Key Concepts

Common Ratio

In a geometric sequence, the term "common ratio" refers to the constant factor that each term is multiplied by to get the next term in the sequence. This is a crucial characteristic of geometric sequences. For example, in a sequence where the common ratio is 2, each term is double the previous one. Mathematically, if you have a sequence defined with the first term as \(a_1\), and common ratio \(r\), subsequent terms are formed as follows:

• \(a_2 = a_1 \cdot r\)
• \(a_3 = a_2 \cdot r = a_1 \cdot r^2\)
• \(a_4 = a_3 \cdot r = a_1 \cdot r^3\)
• and so on.

Understanding the common ratio is essential when analyzing or constructing geometric sequences, as it governs the entire progression of terms.

Subsequence

A subsequence is formed by selecting elements of a sequence while keeping their order. For example, in the original sequence \(\{a_1, a_2, a_3, a_4, a_5, \ldots \}\), a subsequence of this could be \(\{a_1, a_3, a_5, \ldots \}\). The formation of subsequences is not arbitrary; they must maintain the order of appearance from the original sequence.

Analyzing subsequences involves understanding how they inherit properties from the main sequence. For instance, the subsequence \(\{a_1, a_3, a_5, \ldots \}\) still reflects the geometric structure of the original sequence but with modified parameters. The challenge often lies in deducing these properties, such as identifying an independent common ratio for the subsequence.

Mathematical Proof

Mathematical proofs provide a logical framework that explains why a particular property or structure holds. In the context of a geometric sequence and its subsequences, a mathematical proof can confirm whether the subsequence is also geometric.

The solution to whether \(\{a_1, a_3, a_5, \ldots \}\) is geometric starts by showing all terms are expressible in terms of the same power series involving base terms. Then, it calculates the ratio between successive terms to ensure it's constant. Specifically, the proof calculates \(\frac{a_3}{a_1} = r^2\) and \(\frac{a_5}{a_3} = r^2\), confirming they share \(r^2\) as a common ratio, thus validating the subsequence is geometric. This rigorous approach ensures each step logically follows from prior conclusions, making it reliable and verifiable.

Algebraic Expressions

Algebraic expressions play a critical role in understanding and manipulating geometric sequences. They allow for a symbolic representation of sequences and their forms. For geometric sequences, expressions like \(a_n = a_1 \cdot r^{(n-1)}\) are used, where \(a_n\) stands for the nth term, \(a_1\) is the first term, and \(r\) is the common ratio.

Algebraic manipulation helps in deriving terms, such as finding the subsequence \(\{a_1, a_3, a_5, \ldots \}\). In this subsequence, the terms can be rewritten as \(a_1 = a_1\), \(a_3 = a_1 \cdot r^2\), \(a_5 = a_1 \cdot r^4\), showcasing repeated powers of \(r\) linked by specific gaps. Thus, algebraic expressions provide the backbone for formulaic reasoning and problem-solving within the conceptual framework of sequences.