Respuesta :
Given:
The objective is to choose the correct geometric sequence all that apply.
Geometric ratio is defined as the sequence of series containing equal ratio for successive terms.
[tex]r=\frac{a_2}{a_1}=\frac{a_3}{a_2}=\frac{a_4}{a_3}[/tex]Consider option (a), (2, -10, 50, -250, .......).
Here, a1 = 2, a2 = -10, a3 = 50 and a4 = -250
Then the common ratio will be,
[tex]\begin{gathered} r=\frac{-10}{2}=\frac{50}{-10}=\frac{-250}{50} \\ r=-5 \end{gathered}[/tex]Thus, option (a) is a geometric sequence.
Consider option (b) (1, -1, 1, -1, 1...)
Here, a1 = 1, a2 = -1, a3 = 1, a4 = -1.
Then the common ratio will be,
[tex]\begin{gathered} r=\frac{-1}{1}=\frac{1}{-1}=\frac{-1}{1} \\ r=-1 \end{gathered}[/tex]Thus, option (b) is a geometric sequence.
Consider option (c) (0, 10, 20, 30....)
Here, a1 = 0, a2 = 10, a3 = 20, a4 = 30............
Then, the common ratio will be,
[tex]\begin{gathered} r=\frac{10}{0}=\frac{20}{10}=\frac{30}{20} \\ r=\infty\ne2\ne1.5 \end{gathered}[/tex]Thus, option (c) is not a geometric sequence.
Consider option (d) (2.5, 4, 5.5, 7....)
Here, a1 = 2.5, a2 = 4, a3 =5.5 and a4 = 7.
Then, the common ratio will be,
[tex]\begin{gathered} r=\frac{4}{2.5}=\frac{5.5}{4}=\frac{7}{5.5} \\ r=1.6\ne1.3\ne1.27 \end{gathered}[/tex]Thus, option (d) is not a geometric sequence.
Consider option (e) (-13, -6, 1, 8....)
Here, a1 = -13, a2 = -6, a3=1 and a4 = 8.
Then, the common ratio will be,
[tex]\begin{gathered} r=\frac{-6}{-13}=\frac{1}{-6}=\frac{8}{1} \\ r=0.4\ne-0.16\ne8 \end{gathered}[/tex]Thus, option (e) is not a geometric sequence.
Consider option (f), (108, 36, 12, 4...)
Here, a1 = 108, a2 = 36, a3 = 12 and a4 = 4.
Then, the common ratio will be,
[tex]\begin{gathered} r=\frac{36}{108}=\frac{12}{36}=\frac{4}{12} \\ r=\frac{1}{3}=\frac{1}{3}=\frac{1}{3} \end{gathered}[/tex]Thus, option (f) is a geometric sequence.
Hence, the correct geometric sequences are option (a), (b) and (f).
