Problem 35 - Precalculus: Mathematics for Calculus

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: \(a_n = 3^{(2n+1)/3}\).

Step 1: Determine the Common Ratio

A geometric sequence can be expressed in the form \( a, ar, ar^2, ar^3, \ldots \) where \( r \) is the common ratio. To find \( r \), divide the second term by the first term: \( r = \frac{3^{5/3}}{3} = 3^{5/3 - 1} = 3^{2/3} \). Thus, the common ratio is \( 3^{2/3} \).

Step 2: Calculate the Fifth Term

The fifth term of a geometric sequence where the first term \( a = 3 \) and the common ratio \( r = 3^{2/3} \) is given by: \( a_5 = ar^4 = 3 \times (3^{2/3})^4 = 3 \times 3^{8/3} = 3^{1+8/3} = 3^{11/3} \). So, the fifth term is \( 3^{11/3} \).

Step 3: Find the General (n-th) Term Formula

The general term of a geometric sequence \( a, ar, ar^2, \ldots \) is given by \( a_n = ar^{n-1} \). For this sequence, \( a = 3 \) and \( r = 3^{2/3} \), so the n-th term is \( a_n = 3 \times (3^{2/3})^{n-1} = 3^{1 + (2/3)(n-1)} \). Simplifying gives \( a_n = 3^{(2n+1)/3} \).

Key Concepts

Common Ratio

In a geometric sequence, each term after the first is derived by multiplying the previous term by a constant, known as the common ratio. It's crucial to determine this ratio when analyzing the pattern of the sequence.

To find the common ratio, identify any two consecutive terms and divide the latter by the former. For example, in the sequence given in the exercise: 3, \(3^{5/3}\), \(3^{7/3}\), and 27, the common ratio is found as follows:

• Choose the second term: \(3^{5/3}\)
• Divide by the first term: \(3\)

Calculating, we have: \( r = \frac{3^{5/3}}{3} = 3^{5/3 - 3/3} = 3^{2/3} \).

Thus, the common ratio for this sequence is \(3^{2/3}\). This ratio is key in determining further terms in the sequence.

Fifth Term

The fifth term in a geometric sequence is another critical element to identify. It allows us to see how the sequence progresses beyond the initial few terms.

To find the fifth term, we use the formula for any term in a geometric sequence, which is \(a_n = ar^{n-1}\). For our exercise, the first term \(a\) is 3, and the common ratio \(r\) is \(3^{2/3}\).

• The general formula for the fifth term is \(a_5 = ar^4\).
• Substitute the values: \(a_5 = 3 \times (3^{2/3})^4\).
• This simplifies to: \(a_5 = 3 \times 3^{8/3}\).
• Finally, express it as \(a_5 = 3^{1 + 8/3} = 3^{11/3}\).

Thus, the fifth term of this sequence is \(3^{11/3}\). Calculating terms like this helps in visualizing the sequence's behavior as it extends.

n-th Term

The n-th term formula enables us to locate any term in the sequence without listing all preceding terms. This formula is foundational in predicting and understanding the pattern of a geometric sequence.

The general form of the n-th term of a geometric sequence is given by \(a_n = ar^{n-1}\). For our sequence:

• The first term \(a\) is 3.
• The common ratio \(r\) is \(3^{2/3}\).

Hence, the n-th term can be expressed as: \(a_n = 3 \times (3^{2/3})^{n-1}\).

• Simplifying this gives us: \(a_n = 3^{1 + (2/3)(n-1)}\).
• Further simplification results in: \(a_n = 3^{(2n+1)/3}\).

This formula allows you to find the value of any term in the sequence by simply plugging in the value of \(n\). Understanding and using the n-th term formula is essential when working with the vast world of geometric sequences.