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)=x, n=2k+1,` where k is an integer
B. `a_(n)=f(x) " if " n=2k,` where k is an integer
C. The inverse of `a_(n)` exists for any value of n and m
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")`
`a_(1)=f(x)=(a-x^(m))^(1//m)`
`or a_(2)=f(f(x))=[a-{(a-x^(m))^(1//m)}^(m)]^(1//m)=x`
`or a_(3)=f(f(f(x)))=f(x)`
Obviously, the inverse does not exist when m is even and n is odd.