Problem 14 - Algebra and Trigonometry

Question

9–16 Determine whether the sequence is arithmetic. If it is arithmetic, find the common difference. $$\ln 2, \ln 4, \ln 8, \ln 16, \ldots$$

Step-by-Step Solution

Answer

Yes, the sequence is arithmetic with a common difference of \(\ln 2\).

Step 1: Understand Arithmetic Sequence

An arithmetic sequence is a list of numbers where the difference between consecutive terms is constant. This difference is called the common difference, denoted usually by \(d\).

Step 2: Examine Consecutive Terms

The sequence provided is \(\ln 2, \ln 4, \ln 8, \ln 16, \ldots\). We need to check if the difference between consecutive terms is the same.

Step 3: Calculate First Difference

Calculate the difference between the first and second terms: \( \ln 4 - \ln 2 = \ln \left( \frac{4}{2} \right) = \ln 2 \).

Step 4: Calculate Second Difference

Calculate the difference between the second and third terms: \( \ln 8 - \ln 4 = \ln \left( \frac{8}{4} \right) = \ln 2 \).

Step 5: Calculate Third Difference

Calculate the difference between the third and fourth terms: \( \ln 16 - \ln 8 = \ln \left( \frac{16}{8} \right) = \ln 2 \).

Step 6: Determine Consistency and Conclusion

All differences calculated in steps 3, 4, and 5 are equal to \(\ln 2\). This means the sequence is arithmetic with a common difference of \(d = \ln 2\).

Key Concepts

Understanding Common Difference

In an arithmetic sequence, every term is separated by a constant amount, which is called the common difference. This concept is key in identifying whether a series of numbers forms an arithmetic sequence or not. For this, you just have to subtract any term from the subsequent term to get the value of the common difference.

• If the difference remains the same for all consecutive terms, then your sequence is arithmetic.
• Simply denote this common difference as \(d\).

Let’s discuss applying this to the sequence \(\ln 2, \ln 4, \ln 8, \ln 16, \ldots\). Determining the common difference here involves calculating differences using the logarithmic properties. As we compute each difference and all come out as \(\ln 2\), you confirm the sequence as arithmetic with a common difference \(d = \ln 2\).
This means every new term in the sequence can be generated by adding \(\ln 2\) to the previous term.

Applying Logarithms in Sequences

Logarithms can sometimes simplify sequences, making complex relationships more visible. They are especially useful in sequences that involve exponential growth, as they help reduce the exponential terms to linear relationships.

• Using properties of logarithms, such as \(\ln(ab) = \ln a + \ln b\) and \(\ln(a/b) = \ln a - \ln b\), can ease calculation of differences.
• These properties help break down multiplication and division into simple addition and subtraction.

In the sequence \(\ln 2, \ln 4, \ln 8, \ln 16, \ldots\), each term can be seen as \(\ln(2^n)\), which simplifies well using logarithms. This breakdown shows how we apply logarithmic properties to easily identify the common difference \(\ln 2\) in such problems.

Sequence Analysis Techniques

To analyze sequences, whether arithmetic or otherwise, foundational techniques are applied to determine the pattern and establish the type of sequence.

• First, list down the sequence.
• Calculate the difference or ratio between consecutive terms depending on the suspected type.
• Check for consistency in differences or ratios across terms.

In the exercise above, plotting the sequence \(\ln 2, \ln 4, \ln 8, \ln 16, \ldots\), made it essential to compute each difference to confirm repetitive patterns. By verifying the consistent logarithmic difference, you're conducting effective sequence analysis, confirming the sequence as arithmetic. Sequence analysis is not just about number crunching; it’s about recognizing patterns and verifying logical mathematical properties, making it a crucial skill in mathematics.