Answer:
Step-by-step explanation:
[tex]\text{If}\ a,\ b,\ c,\ d,\ e,\ ...\ \text{is a geometric sequence, then}\ \dfrac{b}{a}=\dfrac{c}{b}=\dfrac{d}{c}=\dfrac{e}{d}=....\\\\\#1\\\\\dfrac{7.5}{10}=0.75\\\dfrac{5.625}{7.5}=0.75\\\dfrac{4.21875}{5.625}=0.75\\\boxed{YES}\\================================\\[/tex]
[tex]\#2\\\\\dfrac{40}{160}=\dfrac{1}{4}\\\dfrac{10}{40}=\dfrac{1}{4}\\\dfrac{2.5}{10}=\dfrac{1}{4}\\\boxed{YES}\\================================\\[/tex]
[tex]\#3\\\\\dfrac{70}{20}=3.5\\\dfrac{245}{70}=3.5\\\dfrac{857.5}{245}=3.5\\\boxed{YES}\\================================\\[/tex]
[tex]\#4\\\\\dfrac{16.5}{13}=\dfrac{165}{130}=\dfrac{33}{26}\\\dfrac{20}{16.5}=\dfrac{200}{165}=\dfrac{40}{33}\neq\dfrac{33}{26}\\\boxed{NO}\\================================\\[/tex]
[tex]\#5\\\\\dfrac{5.5}{5}=1.1\\\dfrac{6.05}{5.5}=1.1\\\dfrac{6.655}{6.05}=1.1\\\boxed{YES}\\================================\\[/tex]
[tex]\#6\\\\\dfrac{17.1}{16}=1.06875\\\dfrac{18.2}{17.1}=1.0643274..\neq1.06875\\\boxed{NO}[/tex]
The sequences are geometric,
10, 7.5, 5.625, 4.28175,
160, 40, 10, 2.5,
20, 70, 245, 857.5,
20, 70, 245, 857.5
5, 5.5, 6.05, 6.655
We have to determine, which sequences of the following series are geometric,
According to the question,
A geometric sequence is a sequence in which any element after the first is obtained by multiplying the preceding element by a constant called the common ratio which is denoted by r.
To find the geometric series by using a formula follow all the steps given below.
Here,
If a, b, c, d, e ............ are in geometric progression.
Then,
[tex]\dfrac{b}{a} = \dfrac{c}{b} = \dfrac{d}{c}[/tex]
The series is geometric.
Then,
[tex]\dfrac{b}{a} = \dfrac{c}{b} = \dfrac{d}{c}\\\\ \dfrac{7.5}{10} = \dfrac{5.625}{7.5} = \dfrac{4.21875}{5.625}\\\\0.75 = 0.75 = 0.75[/tex]
Therefore, The series is geometric.
Then,
[tex]\dfrac{b}{a} = \dfrac{c}{b} = \dfrac{d}{c}\\\\ \dfrac{40}{160} = \dfrac{10}{40} = \dfrac{10}{2.5}\\\\ \dfrac{1}{4}= \dfrac{1}{4}= \dfrac{1}{4}[/tex]
Therefore, The series is geometric.
Then,
[tex]\dfrac{b}{a} = \dfrac{c}{b} = \dfrac{d}{c}\\\\ \dfrac{70}{20} = \dfrac{245}{70} = \dfrac{857.5}{245}\\\\ 3.5 = 3.5 = 3.5[/tex]
Therefore, The series is geometric.
Then,
[tex]\dfrac{b}{a} = \dfrac{c}{b} = \dfrac{d}{c}\\\\ \dfrac{13}{16.5} = \dfrac{20}{13} = \dfrac{23.5}{20}\\\\ \dfrac{33}{26}\neq \dfrac{40}{33}\neq \dfrac{47}{40}[/tex]
Therefore, The series is not geometric.
Then,
[tex]\dfrac{b}{a} = \dfrac{c}{b} = \dfrac{d}{c}\\\\ \dfrac{5.5}{5} = \dfrac{6.05}{5.5} = \dfrac{6.655}{6.05}\\\\ 1.1=1.1=1.1[/tex]
Therefore, The series is geometric.
Then,
[tex]\dfrac{b}{a} = \dfrac{c}{b} = \dfrac{d}{c}\\\\ \dfrac{17.1}{16} = \dfrac{18.2}{17.1} = \dfrac{19.3}{18.2}\\\\ 1.068 \neq 1.062 \neq 1.079[/tex]
Therefore, The series not is geometric.
To know more about the Series click the link given below.
https://brainly.com/question/12431044
