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Answer :
To find the common ratio of the sequence [tex]\(-125,-25,-5,-1, \ldots\)[/tex], we need to determine how each term relates to the one before it by multiplication. In a geometric sequence, this relation is called the common ratio.
Here’s how you can find the common ratio:
1. Look at the first two terms of the sequence: [tex]\(-125\)[/tex] and [tex]\(-25\)[/tex].
2. Divide the second term by the first term to find the common ratio.
[tex]\[
\text{Common Ratio} = \frac{\text{Second Term}}{\text{First Term}} = \frac{-25}{-125}
\][/tex]
3. Simplify the fraction:
[tex]\[
\frac{-25}{-125} = \frac{25}{125} = \frac{1}{5}
\][/tex]
So, the common ratio of the sequence is [tex]\(\frac{1}{5}\)[/tex]. This means that each term in the sequence is obtained by multiplying the previous term by [tex]\(\frac{1}{5}\)[/tex].
Here’s how you can find the common ratio:
1. Look at the first two terms of the sequence: [tex]\(-125\)[/tex] and [tex]\(-25\)[/tex].
2. Divide the second term by the first term to find the common ratio.
[tex]\[
\text{Common Ratio} = \frac{\text{Second Term}}{\text{First Term}} = \frac{-25}{-125}
\][/tex]
3. Simplify the fraction:
[tex]\[
\frac{-25}{-125} = \frac{25}{125} = \frac{1}{5}
\][/tex]
So, the common ratio of the sequence is [tex]\(\frac{1}{5}\)[/tex]. This means that each term in the sequence is obtained by multiplying the previous term by [tex]\(\frac{1}{5}\)[/tex].
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