Let
[tex] a1=768\\a2=480\\a3=300\\a4=187.5 [/tex]
Step [tex] 1 [/tex]
Find [tex] \frac{a2}{a1} [/tex]
[tex] \frac{a2}{a1} =\frac{480}{768} [/tex]
Divide by [tex] 96 [/tex] numerator and denominator
[tex] \frac{480}{768} =\frac{\frac{480}{96}}{\frac{768}{96}} \\ \\ \frac{480}{768}=\frac{5}{8} [/tex]
[tex] a2=a1*\frac{5}{8} [/tex]
Step [tex] 2 [/tex]
Find [tex] \frac{a3}{a2} [/tex]
[tex] \frac{a3}{a2} =\frac{300}{480} [/tex]
Divide by [tex] 60 [/tex] numerator and denominator
[tex] \frac{300}{480} =\frac{\frac{300}{60}}{\frac{480}{60}} \\ \\ \frac{300}{480}=\frac{5}{8} [/tex]
[tex] a3=a2*\frac{5}{8} [/tex]
Step [tex] 3 [/tex]
Find [tex] \frac{a4}{a3} [/tex]
[tex] a4=187.5 \\ a4=187\frac{1}{2} \\ \\ a4=\frac{375}{2} [/tex]
[tex] \frac{a4}{a3} =\frac{\frac{375}{2}}{300} \\ \\ \frac{a4}{a3} =\frac{375}{600} [/tex]
Divide by [tex] 75 [/tex] numerator and denominator
[tex] \frac{375}{600} =\frac{\frac{375}{75}}{\frac{600}{75}} \\ \\ \frac{375}{600}=\frac{5}{8} [/tex]
[tex] a4=a3*\frac{5}{8} [/tex]
In this problem
The geometric sequence formula is equal to
[tex] a(n+1)=a(n)*\frac{5}{8}\\ [/tex]
For [tex] n\geq 1 [/tex]
therefore
the answer is
the common ratio for the geometric sequence above is [tex] \frac{5}{8} [/tex]
The common ratio of the geometric sequence [tex]768,480,300,187.5, \ldots[/tex] is [tex]\boxed{\frac{5}{8}}.[/tex]
Further Explanation:
If the first term a and the second term ar is known then, the value of [tex]r[/tex] can be obtained as follows,
[tex]\boxed{r = \frac{{{a_2}}}{{{a_1}}}}[/tex]
The nth term of the geometric sequence can be obtained as,
[tex]\boxed{{a_n} = a \times {r^{n - 1}}}[/tex]
Given:
The geometric sequence is [tex]768,480,300,187.5, \ldots.[/tex]
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 [tex]r[/tex] between the second and first term can be obtained as follows.
[tex]\begin{aligned}r&=\frac{{{a_2}}}{{{a_1}}}\\&= \frac{{480}}{{768}}\\&= \frac{5}{8}\\\end{aligned}[/tex]
The common ratio r between the second and third term can be obtained as follows.
[tex]\begin{aligned}r&= \dfrac{{{a_3}}}{{{a_2}}}\\&=\dfrac{{300}}{{480}}\\&= \dfrac{5}{8}\\\end{gathered}[/tex]
Hence, the common ratio of the geometric sequence [tex]768,480,300,187.5, \ldots[/tex] is [tex]\boxed{\frac{5}{8}}.[/tex]
Learn more:
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.
