If `a_(0)=x,a_(n+1)=f(a_(n)), " where " n=0,1,2, …,` then answer the following questions.
If `f(x)=(1)/(1-x),` then which of the following is not true?

Options

A. `a_(n)=(1)/(1-x) " if " n=3k+1`
B. `a_(n)=(x-1)/(x) " if " n=3k+2`
C. `a_(n)=x " if " n=3k`
D. None of these

Answer

D

Explanation

Given `a_(n+1)=f(a_(n))`
Now, `a_(1)=f(a_(0))=f(x)`
`or a_(2)=f(a_(1))=f(f(a_(0)))=fof(x)`
`or a_(n)=(fofofof …f(x))/("n times")`
Now, if `f(x)=(1)/(1-x), fof(x)=(1)/(1-(1)/(1-x))=(x-1)/(x)`
or `fofof(x)=((1)/(1-x)-1)/((1)/(1-x))=x`
`or a_(n)=(fofof ... of(x))/("n times")=(1)/(1-x) " if " n=3k+1`
`=(x-1)/(x) " if " n=3k+2`
`=x " if " n=3k`