Solution:
Concept:
We will first have to calculate the common ratio of the sequence and see the sequence that gives a common ratio of 1/2
From the first option
The formula given in the question is given below as
[tex]\begin{gathered} f(x+1)=\frac{1}{2}f(x) \\ \frac{f(x+1)}{f(x)}=\frac{1}{2} \end{gathered}[/tex]
From the first option,
[tex]\begin{gathered} r=\frac{T_2}{T_1}=\frac{\frac{x}{2}}{x}=\frac{1}{2} \\ r=\frac{T_3}{T_2}=\frac{\frac{x}{4}}{\frac{x}{2}}=\frac{x}{4}\times\frac{2}{x}=\frac{1}{2} \\ r=\frac{T_4}{T_3}=\frac{\frac{x}{6}}{\frac{x}{4}}=\frac{x}{6}\times\frac{4}{x}=\frac{2}{3}(wrong) \end{gathered}[/tex]
Hence,
The first option is wrong
From the second option,
[tex]\begin{gathered} r=\frac{T_2}{T_1}=\frac{2x}{x}=2 \\ r=\frac{T_3}{T_2}=\frac{4x}{2x}=2 \\ r=\frac{T_4}{T_3}=\frac{8x}{4x}=2(\text{wrong)} \end{gathered}[/tex]
Hence,
The second option is wrong
From the third option,
[tex]\begin{gathered} r=\frac{T_2}{T_1}=\frac{\frac{x}{2}}{x}=\frac{1}{2} \\ r=\frac{T_3}{T_2}=\frac{\frac{x}{4}}{\frac{x}{2}}=\frac{x}{4}\times\frac{2}{x}=\frac{1}{2} \\ r=\frac{T_4}{T_3}=\frac{\frac{x}{8}}{\frac{x}{4}}=\frac{x}{8}\times\frac{4}{x}=\frac{1}{2}(correct) \end{gathered}[/tex]
Hence,
The right answer is the third option
[tex]x,\frac{x}{2},\frac{x}{4},\frac{x}{8}[/tex]
