Problem 89
Question
Explain why the sequence \(\log 2, \log 4\) \(\log 8, \log 16, \ldots\) is an arithmetic sequence.
Step-by-Step Solution
Verified Answer
The sequence has a common difference of \( \log 2 \), so it's arithmetic.
1Step 1: Define General Term of the Sequence
The sequence given is \( \log 2, \log 4, \log 8, \log 16, \ldots \). To find a general term, note that each term can be expressed as a logarithm of a power of 2. Specifically, the \( n \)-th term is \( \log (2^n) \).
2Step 2: Simplify the General Term
Using the properties of logarithms, we can simplify the general form: \( \log (2^n) = n \log 2 \). Thus, the sequence can be rewritten as \( \log 2, 2\log 2, 3\log 2, 4\log 2, \ldots \).
3Step 3: Identify the Common Difference
An arithmetic sequence is defined by having a constant difference between consecutive terms. In our sequence, the difference between consecutive terms is \((n+1)\log 2 - n\log 2 = \log 2\). Thus, the common difference \(d\) is \( \log 2 \).
4Step 4: Conclude that the Sequence is Arithmetic
Since there is a constant difference (\( \log 2 \)) between consecutive terms, the sequence is indeed an arithmetic sequence. Every term is obtained from the previous one by adding \( \log 2 \).
Key Concepts
Logarithm PropertiesCommon DifferenceSequence Terms
Logarithm Properties
Logarithms are quite useful in simplifying complex expressions. One of the key properties we often use is the power rule for logarithms, which states that \( \log(a^b) = b \cdot \log(a) \). This means, anytime you are dealing with a logarithm of a power, you can bring the exponent down in front as a multiplier.
In our sequence \( \log 2, \log 4, \log 8, \log 16, \ldots \), each term can be expressed as \( \log (2^n) \). By applying the power rule, each term transforms to \( n \cdot \log 2 \).
In essence, this property helps turn multiplying numbers inside a log into a simpler multiplication outside. It reduces potential complexity and makes the arithmetic sequence easier to identify.
In our sequence \( \log 2, \log 4, \log 8, \log 16, \ldots \), each term can be expressed as \( \log (2^n) \). By applying the power rule, each term transforms to \( n \cdot \log 2 \).
In essence, this property helps turn multiplying numbers inside a log into a simpler multiplication outside. It reduces potential complexity and makes the arithmetic sequence easier to identify.
Common Difference
The common difference is a foundational concept for arithmetic sequences. It describes the constant amount we add to each term to get the next term.
In our sequence \( \log 2, 2\log 2, 3\log 2, 4\log 2, \ldots \), each term is obtained from the previous term by adding \( \log 2 \). This means:
Identifying this difference is essential because it confirms that the sequence is arithmetic. It creates a predictable pattern for any arithmetic sequence, making it easier to determine future terms or analyze its structure.
In our sequence \( \log 2, 2\log 2, 3\log 2, 4\log 2, \ldots \), each term is obtained from the previous term by adding \( \log 2 \). This means:
- From \( \log 2 \) to \( 2\log 2 \), we add \( \log 2 \).
- From \( 2\log 2 \) to \( 3\log 2 \), we again add \( \log 2 \).
- This pattern continues throughout the sequence.
Identifying this difference is essential because it confirms that the sequence is arithmetic. It creates a predictable pattern for any arithmetic sequence, making it easier to determine future terms or analyze its structure.
Sequence Terms
Understanding sequence terms starts with recognizing how each term fits within the overall sequence.
In this arithmetic sequence \( \log 2, 2\log 2, 3\log 2, 4\log 2, \ldots \), each term follows a specific pattern. They can be represented by the formula \( n\log 2 \), where \( n \) is the term number in the sequence starting from 1. This pattern simplifies predicting any term's value without needing to list every single term.
Consider some of the sequence terms:
In this arithmetic sequence \( \log 2, 2\log 2, 3\log 2, 4\log 2, \ldots \), each term follows a specific pattern. They can be represented by the formula \( n\log 2 \), where \( n \) is the term number in the sequence starting from 1. This pattern simplifies predicting any term's value without needing to list every single term.
Consider some of the sequence terms:
- The first term is \( \log 2 \), which is \( 1 \cdot \log 2 \).
- The second term is \( 2\log 2 \), which follows naturally by the pattern.
- Any \( n \)-th term follows the expression \( n\log 2 \).
Other exercises in this chapter
Problem 88
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