To determine the common ratio of the given geometric sequence [tex]\(\frac{16}{3}\)[/tex], 4, 3, \ldots, let's start by recalling that the ratio between consecutive terms in a geometric sequence is constant.
1. Identify the first few terms:
The first term, [tex]\(a_1\)[/tex], is [tex]\(\frac{16}{3}\)[/tex].
The second term, [tex]\(a_2\)[/tex], is 4.
The third term, [tex]\(a_3\)[/tex], is 3.
2. Calculate the ratio between the second term and the first term:
[tex]\[
\text{Common ratio}, r = \frac{a_2}{a_1} = \frac{4}{\frac{16}{3}}
\][/tex]
To divide by a fraction, we multiply by its reciprocal:
[tex]\[
r = 4 \times \frac{3}{16} = \frac{12}{16} = \frac{3}{4} = 0.75
\][/tex]
3. Calculate the ratio between the third term and the second term to confirm consistency:
[tex]\[
r = \frac{a_3}{a_2} = \frac{3}{4}
\][/tex]
[tex]\[
r = 0.75
\][/tex]
Both ratios confirm that the common ratio is indeed [tex]\(0.75\)[/tex].
Therefore, the common ratio of the geometric sequence [tex]\(\frac{16}{3}\)[/tex], 4, 3, \ldots is [tex]\(0.75\)[/tex].
