Problem 29 - Algebra and Trigonometry

Question

Determine the common ratio, the fifth term, and the \(n\)th term of the geometric sequence. $$ 3,3^{5 / 3}, 3^{7 / 3}, 27, \dots $$

Step-by-Step Solution

Answer

Common ratio: \( 3^{2/3} \), Fifth term: \( 3^{11/3} \), \( n \)th term: \( 3^{(2n + 1)/3} \)

Step 1: Identify the first term

The first term of the sequence is given as 3.

Step 2: Determine the common ratio

To find the common ratio \( r \), divide the second term by the first term. The second term is \( 3^{5/3} \). \[ r = \frac{3^{5/3}}{3} = 3^{5/3 - 1} = 3^{2/3} \] Thus, the common ratio is \( 3^{2/3} \).

Step 3: Calculate the fifth term

To find the fifth term, use the formula for the \( n \)th term of a geometric sequence: \[ a_n = a_1 \cdot r^{n-1} \]Substitute \( a_1 = 3 \), \( r = 3^{2/3} \), and \( n = 5 \):\[ a_5 = 3 \cdot (3^{2/3})^{4} = 3 \cdot 3^{8/3} = 3^{1 + 8/3} = 3^{11/3} \]The fifth term is \( 3^{11/3} \).

Step 4: Formulate the general \( n \)th term

Using the formula for the \( n \)th term: \[ a_n = a_1 \cdot r^{n-1} \]Substitute \( a_1 = 3 \) and \( r = 3^{2/3} \): \[ a_n = 3 \cdot (3^{2/3})^{n-1} = 3 \cdot 3^{2(n-1)/3} = 3^{1 + 2(n-1)/3} \]Simplifying gives: \[ a_n = 3^{(3 + 2n - 2)/3} = 3^{(2n + 1)/3} \] This is the formula for the \( n \)th term.

Key Concepts

Understanding the Common Ratio

In a geometric sequence, each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This characteristic ratio helps determine how the sequence progresses. To find the common ratio in our given sequence, we need to divide one term by the preceding term. For instance, take the second term, \( 3^{5/3} \), and divide it by the first term, 3:

• \( r = \frac{3^{5/3}}{3} = 3^{5/3 - 1} = 3^{2/3} \)

This shows that the common ratio \( r \) is \( 3^{2/3} \).

The common ratio is crucial because it defines the growth pattern of the sequence. Knowing this allows you to calculate any term in the sequence using the common ratio and the first term.

Finding the n-th Term in a Geometric Sequence

The formula to find the \( n \)-th term of a geometric sequence is absolutely essential. This formula allows you to find any term without having to list all the preceding ones. The formula is:

• \( a_n = a_1 \cdot r^{n-1} \)

Where \( a_n \) is the \( n \)-th term, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.

In the sequence provided, with the common ratio \( 3^{2/3} \), and the first term as 3, the \( n \)-th term becomes:

• \( a_n = 3 \cdot (3^{2/3})^{n-1} = 3^{1 + 2(n-1)/3} \)

Simplifying this gives:

• \( a_n = 3^{(2n + 1)/3} \)

This formula can now be used to find any term in the sequence, no matter how large \( n \) is.

Summarizing a Geometric Series

When talking about a geometric sequence, it's also useful to understand what a geometric series is. A geometric series is the sum of the terms in a geometric sequence. In general, if you have a geometric sequence, its series is the sum of the terms:

The formula for the sum \( S_n \) of a geometric series with \( n \) terms is:

• \( S_n = a_1 \frac{1 - r^n}{1 - r} \)

This formula applies only if the absolute value of the common ratio \( r \) is less than 1. If \( r \) is greater than 1, the series does not converge as \( n \) becomes very large. Instead, it will diverge.

For smaller sequences, or when one just wants a finite number of terms, this formula gives a quick way to find the total sum of the sequence's specified terms. This is particularly useful in real-world applications where understanding the total is more important than each individual term.