The common ratio of the geometric sequence \(768,480,300,187.5, \ldots\) is \(\boxed{\frac{5}{8}}.\)
Further Explanation:
If the first term a and the second term ar is known then, the value of \(r\) can be obtained as follows,
\(\boxed{r = \frac{{{a_2}}}{{{a_1}}}}\)
The nth term of the geometric sequence can be obtained as,
\(\boxed{{a_n} = a \times {r^{n - 1}}}\)
Given:
The geometric sequence is \(768,480,300,187.5, \ldots.\)
Explanation:
The first term of the geometric sequence is 768, second term of the geometric sequence is 480, third term 300 and the fourth geometric sequence is 187.5.
The common ratio \(r\) between the second and first term can be obtained as follows.
\(\begin{aligned}r&=\frac{{{a_2}}}{{{a_1}}}\\&= \frac{{480}}{{768}}\\&= \frac{5}{8}\\\end{aligned}\)
The common ratio r between the second and third term can be obtained as follows.
\(\begin{aligned}r&= \dfrac{{{a_3}}}{{{a_2}}}\\&=\dfrac{{300}}{{480}}\\&= \dfrac{5}{8}\\\end{gathered}\)
Hence, the common ratio of the geometric sequence \(768,480,300,187.5, \ldots\) is \(\boxed{\frac{5}{8}}.\)
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: Geometric progression
Keywords: geometric sequence, fraction, written as, common ratio, first term, second term, sum of geometric sequence, 768, 480, 300, 187.5.
