Problem 18 - Algebra and Trigonometry

Question

\(13-20\) . Determine whether the sequence is arithmetic. If is arithmetic, find the common difference. $$ \ln 2, \ln 4, \ln 8, \ln 16, \ldots $$

Step-by-Step Solution

Answer

The sequence is arithmetic with a common difference of \(\ln 2\).

Step 1: Identify the Sequence Terms

The given sequence is \( \ln 2, \ln 4, \ln 8, \ln 16, \ldots \). We recognize the terms are: \( \ln 2, \ln 4, \ln 8, \ln 16, \cdots \).

Step 2: Determine Pattern Among Terms

Notice that each term seems to be the natural logarithm of powers of 2. Specifically, \( \ln 2 = \ln (2^1), \ln 4 = \ln (2^2), \ln 8 = \ln (2^3), \ln 16 = \ln (2^4), \ldots \).

Step 3: Use Logarithm Properties

Using the property of logarithms \( \ln(a^b) = b \ln a \), rewrite each term to better identify the pattern: \( \ln 2 = 1 \cdot \ln 2, \ln 4 = 2 \cdot \ln 2, \ln 8 = 3 \cdot \ln 2, \ln 16 = 4 \cdot \ln 2 \), and so on.

Step 4: Check for Arithmetic Sequence

In an arithmetic sequence, the difference between consecutive terms is constant. Calculate the difference: \( (2\ln 2) - (1\ln 2) = \ln 2 \); \( (3 \ln 2) - (2 \ln 2) = \ln 2 \); \( (4 \ln 2) - (3 \ln 2) = \ln 2 \). The differences are consistent.

Step 5: Identify Common Difference

Given that all differences are the same, confirm that the sequence is arithmetic with a common difference of \( \ln 2 \).

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithm that uses the mathematical constant \( e \) (approximately 2.718) as the base. In simpler terms, if \( \ln x = y \), then \( e^y = x \). This concept is widely used in calculus and mathematical modeling because \( \ln \) is the inverse operation of exponentiation with base \( e \).

Natural logarithms are commonly found in situations involving continuous growth or decay, such as population growth or radioactive decay. For the sequence \( \ln 2, \ln 4, \ln 8, \ln 16, \ldots \), each term represents the natural logarithm of numbers that are powers of 2. Understanding how to work with \( \ln \) helps in simplifying and solving equations involving exponential growth or transformations.

Common Difference

In sequences, a common difference is what we look for when identifying an arithmetic sequence. It is the constant amount that each term increases or decreases by to get to the next term in the sequence. An arithmetic sequence has a consistent difference between consecutive terms.

For example, in the sequence \( \ln 2, \ln 4, \ln 8, \ln 16, \ldots \), if we rewrite the terms using logarithms, it becomes \( 1 \cdot \ln 2, 2 \cdot \ln 2, 3 \cdot \ln 2, 4 \cdot \ln 2 \), and so on. By calculating one term minus the previous one, we find:

• \( (2 \ln 2) - (1 \ln 2) = \ln 2 \)
• \( (3 \ln 2) - (2 \ln 2) = \ln 2 \)
• \( (4 \ln 2) - (3 \ln 2) = \ln 2 \)

Each time, the difference is \( \ln 2 \). This means the sequence is arithmetic with a common difference of \( \ln 2 \). Recognizing this pattern helps in predicting future terms of the sequence.

Properties of Logarithms

Logarithms have several properties that simplify complex calculations and make it easier to work with exponential and logarithmic equations. These properties are especially useful in transforming expressions so that patterns emerge more clearly.

A few of the essential properties include:

• Product Property: \( \ln(ab) = \ln a + \ln b \)
• Quotient Property: \( \ln\left(\frac{a}{b}\right) = \ln a - \ln b \)
• Power Property: \( \ln(a^b) = b \ln a \)

In the given sequence, the Power Property is pivotal. Each term \( \ln 2, \ln 4, \ln 8, \ln 16 \) can be written in the form \( n \cdot \ln 2 \), where \( n \) is a natural number. This conversion showcases the arithmetic nature of the sequence.

By using these properties, especially the Power Property, patterns like those found in sequences become more visible and manageable, aiding in the identification and solution of such problems effectively.