Answer :
To find the common ratio of a geometric sequence, you need to determine the constant factor that each term is multiplied by to get the next term. Let's break down the steps with the given sequence:
The sequence given is: [tex]\(-164, -82, -41, -20.5, \ldots\)[/tex]
Step-by-Step Solution:
1. Identify the first few terms of the sequence:
- First term ([tex]\(a_1\)[/tex]) = [tex]\(-164\)[/tex]
- Second term ([tex]\(a_2\)[/tex]) = [tex]\(-82\)[/tex]
- Third term ([tex]\(a_3\)[/tex]) = [tex]\(-41\)[/tex]
- Fourth term ([tex]\(a_4\)[/tex]) = [tex]\(-20.5\)[/tex]
2. Calculate the common ratio:
The common ratio ([tex]\(r\)[/tex]) in a geometric sequence is found by dividing any term by the previous term. We will do this for a couple of terms to verify that the ratio is consistent.
- Finding the ratio between the second and first term:
[tex]\[
r = \frac{a_2}{a_1} = \frac{-82}{-164}
\][/tex]
Simplifying this fraction:
[tex]\[
r = 0.5
\][/tex]
- Finding the ratio between the third and second term:
[tex]\[
r = \frac{a_3}{a_2} = \frac{-41}{-82}
\][/tex]
Simplifying this fraction:
[tex]\[
r = 0.5
\][/tex]
- Finding the ratio between the fourth and third term:
[tex]\[
r = \frac{a_4}{a_3} = \frac{-20.5}{-41}
\][/tex]
Simplifying this fraction:
[tex]\[
r = 0.5
\][/tex]
3. Verify the common ratio:
Since the ratio between each consecutive term is the same (0.5), we confirm that the common ratio for the given sequence is [tex]\(0.5\)[/tex].
Therefore, the common ratio of the sequence [tex]\(-164, -82, -41, -20.5, \ldots\)[/tex] is:
[tex]\(\frac{1}{2}\)[/tex]
The sequence given is: [tex]\(-164, -82, -41, -20.5, \ldots\)[/tex]
Step-by-Step Solution:
1. Identify the first few terms of the sequence:
- First term ([tex]\(a_1\)[/tex]) = [tex]\(-164\)[/tex]
- Second term ([tex]\(a_2\)[/tex]) = [tex]\(-82\)[/tex]
- Third term ([tex]\(a_3\)[/tex]) = [tex]\(-41\)[/tex]
- Fourth term ([tex]\(a_4\)[/tex]) = [tex]\(-20.5\)[/tex]
2. Calculate the common ratio:
The common ratio ([tex]\(r\)[/tex]) in a geometric sequence is found by dividing any term by the previous term. We will do this for a couple of terms to verify that the ratio is consistent.
- Finding the ratio between the second and first term:
[tex]\[
r = \frac{a_2}{a_1} = \frac{-82}{-164}
\][/tex]
Simplifying this fraction:
[tex]\[
r = 0.5
\][/tex]
- Finding the ratio between the third and second term:
[tex]\[
r = \frac{a_3}{a_2} = \frac{-41}{-82}
\][/tex]
Simplifying this fraction:
[tex]\[
r = 0.5
\][/tex]
- Finding the ratio between the fourth and third term:
[tex]\[
r = \frac{a_4}{a_3} = \frac{-20.5}{-41}
\][/tex]
Simplifying this fraction:
[tex]\[
r = 0.5
\][/tex]
3. Verify the common ratio:
Since the ratio between each consecutive term is the same (0.5), we confirm that the common ratio for the given sequence is [tex]\(0.5\)[/tex].
Therefore, the common ratio of the sequence [tex]\(-164, -82, -41, -20.5, \ldots\)[/tex] is:
[tex]\(\frac{1}{2}\)[/tex]
