To find the common ratio of the geometric sequence [tex]\(3, 9, 27, 81, \ldots\)[/tex], we need to determine the constant ratio between consecutive terms in the sequence.
Step-by-step process:
1. Identify consecutive terms in the sequence: The first few terms of the sequence are 3, 9, 27, and 81.
2. Choose any two consecutive terms to calculate the ratio:
- Calculate the ratio between the second term (9) and the first term (3):
[tex]\[
\text{ratio}_1 = \frac{9}{3} = 3
\][/tex]
- Calculate the ratio between the third term (27) and the second term (9):
[tex]\[
\text{ratio}_2 = \frac{27}{9} = 3
\][/tex]
- Calculate the ratio between the fourth term (81) and the third term (27):
[tex]\[
\text{ratio}_3 = \frac{81}{27} = 3
\][/tex]
3. Verify consistency:
- All calculated ratios ([tex]\(\text{ratio}_1, \text{ratio}_2, \text{ratio}_3\)[/tex]) are equal to 3.
Since the ratio remains constant and equals 3 for all consecutive terms tested, we conclude that the common ratio of the sequence is:
[tex]\[
\boxed{3}
\][/tex]
