Answer:
Option 1 , Option 2 and Option 4 are geometric sequence.
Step-by-step explanation:
A sequence is said to be geometric when their exist a common ratio between the consecutive terms.
Option 1) -2.7,-9,-30,-100...
Common ratio= [tex]r = \frac{a_2}{a_1}[/tex]
= [tex]\frac{-9}{-2.7}[/tex]
= [tex]\frac{10}{3}[/tex]
Common ratio= [tex]r = \frac{a_3}{a_2}[/tex]
= [tex]\frac{-30}{-9}[/tex]
= [tex]\frac{10}{3}[/tex]
Common ratio= [tex]r = \frac{a_4}{a_3}[/tex]
= [tex]\frac{-100}{-30}[/tex]
= [tex]\frac{10}{3}[/tex]
Since the common ratios are equal .
Thus -2.7,-9,-30,-100... is a geometric sequence .
Option 2) -1,2.5, -6.25, 15.625 ...
Common ratio= [tex]r = \frac{a_2}{a_1}[/tex]
= [tex]\frac{2.5}{-1}[/tex]
= [tex]\frac{-5}{2}[/tex]
Common ratio= [tex]r = \frac{a_3}{a_2}[/tex]
= [tex] \frac{-6.25}{2.5}[/tex]
= [tex]\frac{-5}{2}[/tex]
Common ratio= [tex]r = \frac{a_4}{a_3}[/tex]
= [tex]\frac{15.625}{-6.25}[/tex]
= [tex]\frac{-5}{2}[/tex]
Since the common ratios are equal .
Thus -1,2.5, -6.25, 15.625 ... is a geometric sequence .
Option 3)9.1,9.2,9.3,9.4....
Common ratio= [tex]r = \frac{a_2}{a_1}[/tex]
= [tex]\frac{9.2}{9.1}[/tex]
= [tex]1.0109[/tex]
Common ratio= [tex]r = \frac{a_3}{a_2}[/tex]
= [tex]\frac{9.3}{9.2}[/tex]
= [tex]1.0108[/tex]
Common ratio= [tex]r = \frac{a_4}{a_3}[/tex]
= [tex]\frac{9.4}{9.3}[/tex]
= [tex]1.01075[/tex]
Since the common ratios are not equal .
Thus 9.1,9.2,9.3,9.4.... is not a geometric sequence .
Option 4) 8,0.8,0.08,0.008 ....
Common ratio= [tex]r = \frac{a_2}{a_1}[/tex]
= [tex]\frac{0.8}{8}[/tex]
= [tex]0.1[/tex]
Common ratio= [tex]r = \frac{a_3}{a_2}[/tex]
= [tex]\frac{0.08}{0.8}[/tex]
= [tex]0.1[/tex]
Common ratio= [tex]r = \frac{a_4}{a_3}[/tex]
= [tex]\frac{0.008}{0.08}[/tex]
= [tex]0.1[/tex]
Since the common ratios are equal .
Thus -18,0.8,0.08,0.008 .... is a geometric sequence .
Option 5) 4,-4,-12,-20...
Common ratio= [tex]r = \frac{a_2}{a_1}[/tex]
= [tex]\frac{-4}{4}[/tex]
= [tex]-1[/tex]
Common ratio= [tex]r = \frac{a_3}{a_2}[/tex]
= [tex]\frac{-12}{-4}[/tex]
= [tex]3[/tex]
Common ratio= [tex]r = \frac{a_4}{a_3}[/tex]
= [tex]\frac{-20}{-12}[/tex]
= [tex]\frac{5}{3}[/tex]
Since the common ratios are not equal .
Thus 4,-4,-12,-20... is not a geometric sequence .
