The options A, B and C are geometric sequences whereas option D is not.
Step-by-step explanation:
Step 1:
For a series to be a geometric sequence, the numbers in the series must be of a common multiplying ratio. So multiplying a constant value with any number will give the value of the next number.
The multiplying constant can be determined by;
[tex]\frac{2^{nd} term}{1^{st} term} = \frac{3^{rd} term}{2^{nd} term} = \frac{4^{th} term}{3^{rd} term} =[/tex] The common multiplying ratio.
Step 2:
For option A, we determine the multiplying ratio,
[tex]\frac{2^{nd} term}{1^{st} term} = \frac{10}{5} =2,[/tex] [tex]\frac{3^{rd} term}{2^{nd} term} = \frac{20}{10} =2, and[/tex] [tex]\frac{4^{th} term}{3^{rd} term} = \frac{40}{20} =2.[/tex]
Since there is a common multiplying ratio of 2, option A is a geometric series.
Step 3:
For option B, we determine the multiplying ratio,
[tex]\frac{2^{nd} term}{1^{st} term} = \frac{5}{10} =0.5,[/tex] [tex]\frac{3^{rd} term}{2^{nd} term} = \frac{2.5}{5} =0.5, and[/tex] [tex]\frac{4^{th} term}{3^{rd} term} = \frac{2.5}{1.25} =0.5.[/tex]
Since there is a common multiplying ratio of 0.5, option B is also a geometric series.
Step 4:
For option C, we determine the multiplying ratio,
[tex]\frac{2^{nd} term}{1^{st} term} = \frac{3}{1} =3,[/tex] [tex]\frac{3^{rd} term}{2^{nd} term} = \frac{9}{3} =3, and[/tex] [tex]\frac{4^{th} term}{3^{rd} term} = \frac{27}{9} =3.[/tex]
Since there is a common multiplying ratio of 3, option C is a geometric series.
Step 5:
For option D, we determine the multiplying ratio,
[tex]\frac{2^{nd} term}{1^{st} term} = \frac{6}{3} =2,[/tex] [tex]\frac{3^{rd} term}{2^{nd} term} = \frac{9}{6} =1.5, and[/tex] [tex]\frac{4^{th} term}{3^{rd} term} = \frac{12}{9} =1.33.[/tex]
Since there is no common multiplying ratio, option D is not a geometric series.
