To determine the common ratio of the given geometric sequence [tex]\(10, 15, 22.5, 33.75, \ldots \)[/tex], we need to use the property that the ratio between any two consecutive terms in a geometric sequence is constant.
Let's find the ratio between the first and second terms:
[tex]\[
\text{First term (a)} = 10
\][/tex]
[tex]\[
\text{Second term (r \times a)} = 15
\][/tex]
To find the common ratio [tex]\(r\)[/tex]:
[tex]\[
r = \frac{\text{Second term}}{\text{First term}} = \frac{15}{10} = 1.5
\][/tex]
We can verify this ratio by examining the ratio between the second and third terms:
[tex]\[
\text{Third term (r}^2 \times a\text{)} = 22.5
\][/tex]
To find the ratio:
[tex]\[
r = \frac{\text{Third term}}{\text{Second term}} = \frac{22.5}{15} = 1.5
\][/tex]
Thus, the common ratio remains constant at 1.5.
We can further verify with the next term in the sequence as a consistency check:
[tex]\[
\text{Fourth term (r}^3 \times a\text{)} = 33.75
\][/tex]
To find the ratio:
[tex]\[
r = \frac{\text{Fourth term}}{\text{Third term}} = \frac{33.75}{22.5} = 1.5
\][/tex]
The common ratio of the sequence is indeed constant and is [tex]\(1.5\)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{1.5}
\][/tex]
