Problem 42 - Precalculus with Limits

Question

In Exercises 35 - 44, write an expression for the \( n \)th term of the geometric sequence. Then find the indicated term. \( a_1 = 1, r = \sqrt{3}, n = 8 \)

Step-by-Step Solution

Answer

The expression for the \(n\)-th term of the geometric sequence is \(a_n = (\sqrt{3})^{(n-1)}\), and the 8th term is \((\sqrt{3})^{7}\).

Step 1: Write down the given parameters of the geometric sequence

From the exercise we extract the first term, \(a_1 = 1\), ratio, \(r = \sqrt{3}\), and that we are looking for the 8th term, \(n = 8\).

Step 2: Formulate the n-th term expression of the geometric sequence

The general expression for the \(n\)-th term of a geometric sequence is, \(a_n = a_1*r^{(n-1)}\). Here, substituting the given values, we get that formula to be \(a_n = 1* (\sqrt{3})^{(n-1)}\) or simply \(a_n = (\sqrt{3})^{(n-1)}\).

Step 3: Find the 8th term of the sequence.

Substitute \(n = 8\) into the expression to find the 8th term: \(a_8 = (\sqrt{3})^{(8-1)} = (\sqrt{3})^{7}\).

Key Concepts

n-th term formula

The n-th term formula in a geometric sequence is key to determining any specific term within the sequence. A geometric sequence is a list of numbers where each number after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The formula to find the n-th term, denoted as \( a_n \), is given by:
\[ a_n = a_1 \times r^{(n-1)} \]where:

• \( a_1 \) is the first term of the sequence.
• \( r \) is the common ratio.
• \( n \) is the term number we want to find.

To find any term in the sequence, just substitute the values for \( a_1 \), \( r \), and \( n \) into the formula, and you'll get the term you need. This straightforward approach makes it easy to access any term without calculating all prior terms.

common ratio

The concept of the common ratio is fundamental in understanding geometric sequences. The common ratio \( r \) is the factor by which each term in the sequence is multiplied to get the next term. It's constant throughout the sequence, which is what distinguishes a geometric sequence from an arithmetic one.
For example, in a sequence where the first term \( a_1 = 1 \) and the ratio \( r = \sqrt{3} \), each subsequent term is obtained by multiplying the previous term by \( \sqrt{3} \). Knowing the common ratio not only helps in generating all the terms of the sequence but also plays a crucial role in determining the n-th term using the n-th term formula. You can think of the common ratio as the secret ingredient that defines the flavor of the sequence.

sequence analysis

When analyzing a geometric sequence, it’s all about understanding the pattern and behavior of the terms. To start with, identify the sequence's elements: the first term and the common ratio. With these, determine how each term progresses and how they are related through multiplication.
Sequence analysis involves:

• Calculating specific terms using the n-th term formula.
• Investigating properties like whether terms grow exponentially as seen with a ratio greater than 1.
• Exploring convergence or divergence of the sequence depending on the magnitude of the common ratio.

Understanding these properties aids in predicting not just individual terms but also the overall behavior of the sequence over time. This thorough analysis helps make informed conclusions about the sequence dynamics and potential applications.