Problem 9 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 3-10: Find the \(n\)th term, the fifth term, and the tenth term of the arithmetic sequence. $$ \ln 3, \ln 9, \ln 27, \ln 81, \ldots $$

Step-by-Step Solution

Answer

\( a_n = n \cdot \ln 3 \); Fifth term: \( 5 \cdot \ln 3 \); Tenth term: \( 10 \cdot \ln 3 \).

Step 1: Determine the Common Difference

The given sequence is \, \( \ln 3 \), \( \ln 9 \), \( \ln 27 \), \( \ln 81 \), \ldots Identify the common difference between consecutive terms. Calculate: \( \ln 9 - \ln 3 = \ln \left( \frac{9}{3} \right) = \ln 3 \). The common difference \( d \) is \( \ln 3 \).

Step 2: Write the Formula for the \(n\)th Term

An arithmetic sequence's \(n\)th term is given by \( a_n = a_1 + (n-1) \cdot d \).The first term \( a_1 = \ln 3 \) and the common difference \( d = \ln 3 \).Thus, the formula becomes \( a_n = \ln 3 + (n-1) \cdot \ln 3 = n \cdot \ln 3 \).

Step 3: Find the Fifth Term

Substitute \( n = 5 \) into the \(n\)th term formula: \( a_5 = 5 \cdot \ln 3 \).

Step 4: Find the Tenth Term

Substitute \( n = 10 \) into the \(n\)th term formula: \( a_{10} = 10 \cdot \ln 3 \).

Key Concepts

Common Difference

An arithmetic sequence is a series of numbers in which the difference between any two successive members is constant. This consistent difference is known as the common difference. In our exercise involving natural logarithms of powers of 3, the sequence given is \( \ln 3, \ln 9, \ln 27, \ln 81, \ldots \). To find the common difference \( d \), we subtract the logarithm of the first term from the logarithm of the second term:

• \( \ln 9 - \ln 3 = \ln \left( \frac{9}{3} \right) = \ln 3 \)

This tells us that each term in the sequence increases by \( \ln 3 \), making it the common difference. Understanding this concept is crucial as it enables us to predict any term in the sequence by using the difference and the first term.

Nth Term Formula

The formula for finding the \(n\)th term of an arithmetic sequence is important because it allows calculation of any term in the sequence without listing every single term. For an arithmetic sequence, this formula is given by:

• \( a_n = a_1 + (n-1) \cdot d \)

where : - \( a_1 \) is the first term. - \( d \) is the common difference. In the exercise, \( a_1 = \ln 3 \) and \( d = \ln 3 \), so the formula becomes:

• \( a_n = \ln 3 + (n-1) \cdot \ln 3 = n \cdot \ln 3 \)

This formula provides a straightforward method to identify any term in the sequence by plugging in the desired value of \( n \).

Logarithms

Logarithms are mathematical operations that are the reverse of exponentiation. The notation \( \ln \) specifically refers to the natural logarithm, which has a base of \( e \) (approximately 2.718). When dealing with sequences that involve logarithms, like the one in this exercise, understanding properties of logarithms can be very helpful. For instance, the property \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \) is instrumental in simplifying and identifying the common difference in logarithmic sequences. Logarithms can help turn multiplicative processes into additive ones, which simplifies many calculations and makes solving problems related to sequences of geometric growth more manageable.

Sequence Terms Calculation

Calculating specific terms in an arithmetic sequence once you have the \(n\)th term formula is straightforward. Here's how you can find specific terms using the formula \( a_n = n \cdot \ln 3 \):

• To find the fifth term, we set \( n = 5 \). Then \( a_5 = 5 \cdot \ln 3 \), which gives us the value of the fifth term.
• To find the tenth term, set \( n = 10 \). Thus, \( a_{10} = 10 \cdot \ln 3 \), yielding the tenth term.

These calculations underline the importance of understanding the sequence structure and using it to derive specific information without recalculating each sequence element manually. This process is both efficient and reduces the risk of errors.