To determine the common ratio of the given geometric sequence [tex]\(6, 15, \frac{75}{2}, \frac{375}{4}, \ldots\)[/tex], we follow these steps:
1. Identify the first few terms of the sequence:
- The first term [tex]\(a_1\)[/tex] is [tex]\(6\)[/tex].
- The second term [tex]\(a_2\)[/tex] is [tex]\(15\)[/tex].
- The third term [tex]\(a_3\)[/tex] is [tex]\(\frac{75}{2}\)[/tex].
- The fourth term [tex]\(a_4\)[/tex] is [tex]\(\frac{375}{4}\)[/tex].
2. The common ratio [tex]\(r\)[/tex] in a geometric sequence is calculated by dividing any term by its preceding term. We calculate the common ratio between each pair of successive terms:
- For [tex]\(a_2\)[/tex] and [tex]\(a_1\)[/tex]:
[tex]\[
r = \frac{a_2}{a_1} = \frac{15}{6} = 2.5
\][/tex]
- For [tex]\(a_3\)[/tex] and [tex]\(a_2\)[/tex]:
[tex]\[
r = \frac{a_3}{a_2} = \frac{\frac{75}{2}}{15} = \frac{75}{2 \times 15} = \frac{75}{30} = 2.5
\][/tex]
- For [tex]\(a_4\)[/tex] and [tex]\(a_3\)[/tex]:
[tex]\[
r = \frac{a_4}{a_3} = \frac{\frac{375}{4}}{\frac{75}{2}} = \frac{375 \times 2}{4 \times 75} = \frac{750}{300} = 2.5
\][/tex]
3. Since the common ratio [tex]\(r\)[/tex] is consistent between each pair of successive terms and equals [tex]\(2.5\)[/tex], we conclude that the common ratio of the given geometric sequence is:
[tex]\[
r = 2.5
\][/tex]
Thus, the option [tex]\( r = \frac{5}{2} \)[/tex] matches the calculated common ratio.
