To find the common ratio of the sequence [tex]\(-2, 6, -18, 54, \ldots\)[/tex] and determine if it is consistent, we'll take the following steps:
1. Identify the first few terms of the sequence:
- First term ([tex]\(a_1\)[/tex]): [tex]\(-2\)[/tex]
- Second term ([tex]\(a_2\)[/tex]): [tex]\(6\)[/tex]
- Third term ([tex]\(a_3\)[/tex]): [tex]\(-18\)[/tex]
- Fourth term ([tex]\(a_4\)[/tex]): [tex]\(54\)[/tex]
2. Calculate the ratios of each consecutive pair of terms to find the common ratio ([tex]\(r\)[/tex]):
[tex]\[
r_1 = \frac{a_2}{a_1} = \frac{6}{-2} = -3
\][/tex]
[tex]\[
r_2 = \frac{a_3}{a_2} = \frac{-18}{6} = -3
\][/tex]
[tex]\[
r_3 = \frac{a_4}{a_3} = \frac{54}{-18} = -3
\][/tex]
3. Check if the calculated ratios are the same for each pair of terms. In this case:
[tex]\[
r_1 = r_2 = r_3 = -3
\][/tex]
Since all the ratios are the same, the common ratio for the given geometric sequence is:
[tex]\[
\boxed{-3}
\][/tex]
