Question

Show that each sequence is arithmetic. Find the common difference, and list the first four terms. $$ \left\\{s_{n}\right\\}=\left\\{\ln 3^{n}\right\\} $$

Short answer

The sequence is arithmetic with a common difference of \(\ln 3\). The first four terms are \(\ln 3, 2 \ln 3, 3 \ln 3, 4 \ln 3\).

Step-by-step solution

- Understand the Sequence

Given the sequence: \ \( \{s_n\} = \{\ln 3^n\} \). The task is to show that the sequence is arithmetic, find the common difference, and list the first four terms.

- Simplify the Expression

Simplify \(s_n\) using logarithmic properties: \ \( \ln 3^n = n \ln 3 \). Thus, \(s_n = n \ln 3\).

- Identify the Common Difference

To prove the sequence is arithmetic, we need to check if the difference between consecutive terms is constant. Calculate the difference: \ \( s_{n+1} - s_n = ((n+1) \ln 3) - (n \ln 3) = \ln 3 \). Hence, the common difference is \( \ln 3 \).

- List the First Four Terms

Calculate the first four terms by plugging in the values of \(n\) (1, 2, 3, 4) into \(s_n = n \ln 3\): \ \( s_1 = 1 \ln 3 \), \( s_2 = 2 \ln 3 \), \( s_3 = 3 \ln 3 \), \( s_4 = 4 \ln 3 \). Therefore, the first four terms are \( \ln 3, 2 \ln 3, 3 \ln 3, 4 \ln 3 \).

Key concepts

common difference

In an arithmetic sequence, the common difference is the amount added to each term to get to the next term. For the given sequence, we simplified it to \(s_n = n \, \ln 3\). To find the common difference, we need to subtract a term from its next term: \(s_{n+1} - s_n = (n + 1) \, \ln 3 - n \, \ln 3\). You can see that this simplifies to \(\ln 3\). This means \(\ln 3\) is the common difference. It's the consistent amount added to each term to get to the next one.

logarithmic properties

Logarithmic properties help simplify and manipulate expressions with logarithms. Here are a few key properties used widely:

• Product Property: \(\log_b(MN) = \log_b(M) + \log_b(N)\)
• Quotient Property: \(\log_b(M/N) = \log_b(M) - \log_b(N)\)
• Power Property: \(\log_b(M^p) = p \log_b(M)\)

For our sequence \(\{ \ln 3^n \}\), we use the Power Property to simplify the term: \(\ln 3^n = n \ln 3\). This makes it easier to work with, showing that the sequence is indeed arithmetic.

arithmetic progression

An arithmetic progression (or arithmetic sequence) is a sequence of numbers where the difference between any two consecutive terms is always the same. This constant difference is known as the common difference. For the sequence \(\{s_n\} = \{\ln 3^n\}\), we transformed it to \(\{s_n\} = \{n \ln 3\}\). Since the difference between consecutive terms \(s_{n+1} - s_n\) is constant (\(\ln 3\)), this confirms that our sequence is arithmetic. It's important to note that arithmetic progressions can help us easily predict future terms in the series by just adding the common difference.

first four terms

To list the first four terms of the sequence, we substitute the first four values of \(n\) into the simplified expression \(s_n = n \ln 3\):

• \(s_1 = 1\ln 3 = \ln 3\)
• \(s_2 = 2\ln 3\)
• \(s_3 = 3\ln 3\)
• \(s_4 = 4\ln 3\)

So, the first four terms are \(\ln 3, 2 \ln 3, 3 \ln 3, 4 \ln 3\). These terms are simply multiples of \(\ln 3\). This reinforces that we are dealing with an arithmetic sequence, given the predictable pattern formed by the common difference.