Question

find the common ratio \\(r\\) for the geometric sequence and use \\(r\\) to find the next three terms.

1. \\(972, 324, 108, 36, \dots\\)

first, identify each term in the sequence. for this sequence,
\\(f(1) = \square\\), \\(f(2) = \square\\), \\(f(3) = \square\\) and \\(f(4) = \square\\)
then, find the common ratio \\(r\\).
\\(r = \frac{\square}{\square} = \square\\)
lastly, use \\(f(n) = f(n - 1) \cdot r\\) with \\(r = \square\\) to find \\(f(5)\\), \\(f(6)\\), and \\(f(7)\\).
\\(f(5) = f(4) \cdot \square\\)
\\(f(5) = \square \cdot \square\\)
\\(f(5) = \square\\)
\\(f(6) = f(5) \cdot \square\\)
\\(f(6) = \square \cdot \square\\)
\\(f(6) = \square\\)
\\(f(7) = f(6) \cdot \square\\)
\\(f(7) = \square \cdot \square\\)
\\(f(7) = \square\\)
the common ratio of the geometric sequence is \\(\underline{\quad\quad}\\) and the next three terms of the sequence are \\(\underline{\quad\quad}\\)

Explanation:

Identify the terms

We identify the first four terms of the given geometric sequence \(972, 324, 108, 36, \dots\) using Geometric Sequences:

• \(f(1) = 972\)
• \(f(2) = 324\)
• \(f(3) = 108\)
• \(f(4) = 36\)

Find the common ratio

We calculate the Common Ratio \(r\) by dividing consecutive terms:

$$ r = \frac{f(2)}{f(1)} = \frac{324}{972} = \frac{1}{3} $$

Calculate the fifth term

Using the recursive formula \(f(n) = f(n-1) \cdot r\) with \(r = \frac{1}{3}\):

• \(f(5) = f(4) \cdot \frac{1}{3}\)
• \(f(5) = 36 \cdot \frac{1}{3}\)
• \(f(5) = 12\)

Calculate the sixth term

Using the recursive formula with \(r = \frac{1}{3}\):

• \(f(6) = f(5) \cdot \frac{1}{3}\)
• \(f(6) = 12 \cdot \frac{1}{3}\)
• \(f(6) = 4\)

Calculate the seventh term

Using the recursive formula with \(r = \frac{1}{3}\):

• \(f(7) = f(6) \cdot \frac{1}{3}\)
• \(f(7) = 4 \cdot \frac{1}{3}\)
• \(f(7) = \frac{4}{3}\)

Answer:

Find the common ratio \(r\) for the geometric sequence and use \(r\) to find the next three terms.

1. \(972, 324, 108, 36, \dots\)

First, identify each term in the sequence. For this sequence,
\(f(1) =\) <blank>972</blank>, \(f(2) =\) <blank>324</blank>, \(f(3) =\) <blank>108</blank>, and \(f(4) =\) <blank>36</blank>.

Then, find the common ratio \(r\).

$$ r = \frac{\text{324}}{\text{972}} = \text{\(\frac{1}{3}\)} $$

Lastly, use \(f(n) = f(n-1) \cdot r\) with \(r =\) <blank>\(\frac{1}{3}\)</blank> to find \(f(5)\), \(f(6)\), and \(f(7)\).

\(f(5) = f(4) \cdot\) <blank>\(\frac{1}{3}\)</blank>
\(f(5) =\) <blank>36</blank> \(\cdot\) <blank>\(\frac{1}{3}\)</blank>
\(f(5) =\) <blank>12</blank>

\(f(6) = f(5) \cdot\) <blank>\(\frac{1}{3}\)</blank>
\(f(6) =\) <blank>12</blank> \(\cdot\) <blank>\(\frac{1}{3}\)</blank>
\(f(6) =\) <blank>4</blank>

\(f(7) = f(6) \cdot\) <blank>\(\frac{1}{3}\)</blank>
\(f(7) =\) <blank>4</blank> \(\cdot\) <blank>\(\frac{1}{3}\)</blank>
\(f(7) =\) <blank>\(\frac{4}{3}\)</blank>

The common ratio of the geometric sequence is <blank>\(\frac{1}{3}\)</blank> and the next three terms of the sequence are <blank>\(12, 4, \frac{4}{3}\)</blank>.