Problem 33 - Algebra and Trigonometry

Question

Determine the common ratio, the fifth term, and the nth 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: \( 3^{11/3} \). nth term: \( 3^{(2n+1)/3} \).

Step 1: Identify the First Term

In the given sequence, the first term \( a_1 \) is \( 3 \).

Step 2: Determine the Common Ratio

To find the common ratio \( r \), use the formula \( r = \frac{a_2}{a_1} \). The second term is \( 3^{5/3} \). Thus, \( r = \frac{3^{5/3}}{3} = 3^{5/3 - 1} = 3^{2/3} \).

Step 3: Find the Fifth Term

The fifth term, \( a_5 \), is found using the formula for the \( n \)-th term of a geometric sequence: \( a_5 = a_1 \cdot r^{4} \). So \( a_5 = 3 \cdot (3^{2/3})^4 = 3 \cdot 3^{8/3} = 3^{1 + 8/3} = 3^{11/3} \).

Step 4: Express the nth Term

The general formula for the \( n \)-th term of a geometric sequence is \( a_n = a_1 \cdot r^{n-1} \). Substituting the values we found, we get \( a_n = 3 \cdot (3^{2/3})^{n-1} = 3 \cdot 3^{2(n-1)/3} = 3^{1 + 2(n-1)/3} = 3^{(2n+1)/3} \).

Key Concepts

Understanding the Common Ratio in a Geometric Sequence

A geometric sequence is a list of numbers where each term, after the first, is found by multiplying the previous number by a fixed, non-zero number called the 'common ratio'. This ratio represents the constant factor between consecutive terms in the sequence.

To determine the common ratio in a sequence, divide the second term by the first term. In the provided sequence, the terms are given as \(3, 3^{5/3}, 3^{7/3}, 27, \ldots\). Here, the first term, \(a_1\), is \(3\), and the second term, \(a_2\), is \(3^{5/3}\).

• Common ratio \(r\) is calculated as \(r = \frac{a_2}{a_1} = \frac{3^{5/3}}{3}\).
• Simplifying this, we find \(r = 3^{5/3 - 1} = 3^{2/3}\).

This simplification shows that our common ratio is \(3^{2/3}\). Having determined the common ratio is crucial because it allows us to find other terms in the sequence with ease.

Exploring the nth Term Formula

The formula for the nth term of a geometric sequence provides a way to find any term in the sequence without listing all the previous terms by utilizing the common ratio. This formula is given as \(a_n = a_1 \cdot r^{n-1}\), where \(a_n\) is the nth term you wish to find, \(a_1\) is the first term, and \(r\) is the common ratio.

In our sequence, we already know that:

• \(a_1 = 3\)
• \(r = 3^{2/3}\)

Substituting these into the nth-term formula gives us \(a_n = 3 \cdot (3^{2/3})^{n-1} = 3 \cdot 3^{2(n-1)/3}\).

Simplifying further, \(a_n = 3^{1 + 2(n-1)/3} = 3^{(2n+1)/3}\). This equation lets you calculate any term in the sequence by just plugging in the value of \(n\).

Calculating the Fifth Term of a Geometric Sequence

To find the fifth term in a geometric sequence using our nth-term formula can clearly demonstrate how all these components (common ratio, nth term formula) come together. By applying the sequence's formula, you can solve for \(a_5\), the fifth term.

Using \(a_5 = 3 \cdot (3^{2/3})^{4}\):

• Firstly, calculate \((3^{2/3})^4 = 3^{8/3}\).
• Then, multiply this by the first term, \(3\).
• This results in \(a_5 = 3 \cdot 3^{8/3} = 3^{1 + 8/3} = 3^{11/3}\).

This shows that the fifth term in this particular sequence is \(3^{11/3}\).

Finding individual terms allows us to see the exponential nature of growth in a geometric sequence.