Question

Question was provided via image upload.

Explanation:

Identify terms in sequence

Using the Geometric Sequences knowledge point

$$
LATEXBLOCK0
$$

Calculate common ratio

Using the Common Ratio knowledge point

$$ r = \frac{20}{5} = 4 $$

Find the fifth term

Using the Geometric Sequences and Common Ratio knowledge points

$$
LATEXBLOCK1
$$

Find the sixth term

Using the Geometric Sequences and Common Ratio knowledge points

$$
LATEXBLOCK2
$$

Find the seventh term

Using the Geometric Sequences and Common Ratio knowledge points

$$
LATEXBLOCK3
$$

Answer:

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

1. \(5, 20, 80, 320, \dots\)

First, identify each term in the sequence. For this sequence,
\(f(1) =\) <blank>5</blank>, \(f(2) =\) <blank>20</blank>, \(f(3) =\) <blank>80</blank>, and \(f(4) =\) <blank>320</blank>.

Then, find the common ratio \(r\).
\(r =\) <blank>\(\frac{20}{5}\)</blank> \(=\) <blank>4</blank>.

Lastly, use \(f(n) = f(n-1) \cdot r\) with \(r =\) <blank>4</blank> to find \(f(5)\), \(f(6)\), and \(f(7)\).

\(f(5) = f(4) \cdot\) <blank>4</blank>
\(f(5) =\) <blank>320</blank> \(\cdot\) <blank>4</blank>
\(f(5) =\) <blank>1280</blank>

\(f(6) = f(5) \cdot\) <blank>4</blank>
\(f(6) =\) <blank>1280</blank> \(\cdot\) <blank>4</blank>
\(f(6) =\) <blank>5120</blank>

\(f(7) = f(6) \cdot\) <blank>4</blank>
\(f(7) =\) <blank>5120</blank> \(\cdot\) <blank>4</blank>
\(f(7) =\) <blank>20480</blank>

The common ratio of the geometric sequence is <blank>4</blank> and the next three terms of the sequence are <blank>1280</blank>, <blank>5120</blank>, and <blank>20480</blank>.