To determine the common ratio in the given sequence [tex]\(5, 15, 45, 135\)[/tex], let's follow these steps:
1. Identify the terms of the sequence:
The first term is [tex]\(5\)[/tex], the second term is [tex]\(15\)[/tex], the third term is [tex]\(45\)[/tex], and the fourth term is [tex]\(135\)[/tex].
2. Calculate the ratio between consecutive terms:
The common ratio [tex]\(r\)[/tex] can be found by dividing any term by the previous term.
Let's calculate the ratio using the first and second term:
[tex]\[
r = \frac{15}{5} = 3
\][/tex]
Next, we will verify this ratio using the second and third term:
[tex]\[
\frac{45}{15} = 3
\][/tex]
Finally, we confirm the consistency of the ratio using the third and fourth term:
[tex]\[
\frac{135}{45} = 3
\][/tex]
3. Conclusion:
Since the ratio between each pair of consecutive terms is consistently [tex]\(3\)[/tex], the common ratio for the sequence is [tex]\(3\)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{3}
\][/tex]
