Problem 24 - Precalculus: Mathematics for Calculus

Question

The first four terms of a sequence are given. Can these terms be the terms of an arithmetic sequence? If so, find the common difference. $$\ln 2, \ln 4, \ln 8, \ln 16, \dots$$

Step-by-Step Solution

Answer

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

Step 1: Understand Arithmetic Sequences

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

Step 2: Review the Sequence Terms

The given sequence is \(\ln 2, \ln 4, \ln 8, \ln 16, \dots\). Each term is the natural logarithm of a number that is a power of 2.

Step 3: Express Terms as Logarithms

Rewrite each term using logarithmic identities: \(\ln 4 = \ln (2^2) = 2 \ln 2\), \(\ln 8 = \ln (2^3) = 3 \ln 2\), \(\ln 16 = \ln (2^4) = 4 \ln 2\).

Step 4: Identify the Sequence Pattern

The sequence can now be expressed as \(\ln 2, 2 \ln 2, 3 \ln 2, 4 \ln 2, \dots\). This is indeed an arithmetic sequence where each consecutive term increases by \(\ln 2\).

Step 5: Calculate the Common Difference

The common difference \(d\) is found by subtracting the first term from the second: \(2 \ln 2 - \ln 2 = \ln 2\). Therefore, the common difference is \(\ln 2\).

Key Concepts

Common Difference

The concept of a common difference is crucial in understanding arithmetic sequences. In an arithmetic sequence, the difference between each consecutive term remains constant. This difference is known as the "common difference" and is usually denoted by the letter \(d\).

When you have a sequence, like the one given in the exercise (\(\ln 2, \ln 4, \ln 8, \ln 16, \dots\)), you determine if it's arithmetic by checking for a common difference.

• Subtract the first term from the second to find \(d\).
• Check if the same \(d\) applies to the subtraction of subsequent terms.

In this context, the common difference was found to be \(\ln 2\), as each term increases by \(\ln 2\) compared to the previous one.

Logarithmic Identities

Logarithmic identities are vital tools when dealing with expressions involving logarithms. These identities help us simplify or rewrite logarithmic expressions to uncover patterns or solve equations. In this exercise, one of the most important identities used is the power identity, which states:

• \(\ln(a^b) = b \cdot \ln(a)\)

This identity simplifies logarithms of powers, which helped transform our sequence terms. In the sequence \(\ln 4, \ln 8, \ln 16\), each term can be rewritten using the power identity:

• \(\ln 4 = 2 \cdot \ln 2\), because \(4 = 2^2\).
• \(\ln 8 = 3 \cdot \ln 2\), because \(8 = 2^3\).
• \(\ln 16 = 4 \cdot \ln 2\), because \(16 = 2^4\).

Applying these identities reveals the arithmetic sequence structure, as each step is a successive multiple of \(\ln 2\).

Logarithmic identities are incredibly useful in simplifying complex expressions and often reveal hidden patterns in sequences.

Natural Logarithm

Natural logarithms are logarithms with the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. The natural logarithm of a number \(x\) is commonly written as \(\ln(x)\).

Natural logarithms are essential in mathematics because they appear in various formulas and are used to solve problems involving growth and decay, such as in finance or natural processes.

• In exponential growth, the natural logarithm helps invert exponential equations.
• In calculus, \(\ln(x)\) has straightforward derivatives and integrals.

In our exercise, each term of the sequence is a natural logarithm of a power of 2. Using this characteristic, we can analyze the sequence with logarithmic identities, identifying patterns, and verifying that the sequence is arithmetic with \(\ln 2\) as the common difference.

Understanding the properties of natural logarithms adds another layer to solving complex problems, making them a powerful mathematical tool.