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Answer :
To determine the exponential function of the given geometric sequence [tex]\(125, 25, 5, 1, \ldots\)[/tex], we follow these steps:
1. Identify the first term of the sequence:
The first term, often denoted as [tex]\( a \)[/tex], is 125.
2. Determine the common ratio:
The common ratio of a geometric sequence can be found by dividing any term by the preceding term.
For this sequence:
[tex]\[
\text{common ratio } r = \frac{25}{125} = \frac{1}{5}
\][/tex]
3. Write the exponential function:
The general formula for the exponential function of a geometric sequence is given by:
[tex]\[
y = a \left(r\right)^x
\][/tex]
Substituting the values we identified:
[tex]\[
y = 125 \left(\frac{1}{5}\right)^x
\][/tex]
This function [tex]\( y = 125 \left(\frac{1}{5}\right)^x \)[/tex] represents the geometric sequence provided. Each term of the sequence can be found by replacing [tex]\( x \)[/tex] with corresponding values (0, 1, 2, ...), where [tex]\( x = 0 \)[/tex] gives 125, for example.
1. Identify the first term of the sequence:
The first term, often denoted as [tex]\( a \)[/tex], is 125.
2. Determine the common ratio:
The common ratio of a geometric sequence can be found by dividing any term by the preceding term.
For this sequence:
[tex]\[
\text{common ratio } r = \frac{25}{125} = \frac{1}{5}
\][/tex]
3. Write the exponential function:
The general formula for the exponential function of a geometric sequence is given by:
[tex]\[
y = a \left(r\right)^x
\][/tex]
Substituting the values we identified:
[tex]\[
y = 125 \left(\frac{1}{5}\right)^x
\][/tex]
This function [tex]\( y = 125 \left(\frac{1}{5}\right)^x \)[/tex] represents the geometric sequence provided. Each term of the sequence can be found by replacing [tex]\( x \)[/tex] with corresponding values (0, 1, 2, ...), where [tex]\( x = 0 \)[/tex] gives 125, for example.
Thank you for reading the article Write the exponential function of the geometric sequence tex 125 25 5 1 ldots tex A tex y 125 5 x tex B tex y. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!
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Rewritten by : Jeany
