If `a_(0)=x,a_(n+1)=f(a_(n)), " where " n=0,1,2, …,` then answer the following questions.
If `f:R to R ` is given by `f(x)=3+4x and a_(n)=A+Bx,` then which of the following is not true?

Options

A. `A+B+1=2^(2n+1)`
B. `|A-B|=1`
C. `underset (h to oo)(lim)(A)/(B)= -1`
D. None of these

Answer

C

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")`
Since `a_(1)=g(x)=3+4x,` we have
`a_(2)=f{g(x)}=g(3+4x)=3+4(3+4x)=(4^(2)-1)+4^(2)x`
`a_(3)=g{g^(2)(x)}=g(15+4^(2)x)=3+4(15+4^(2)x)=63+4^(3)x`
`=(4^(3)-1)+4^(3)x`
Similarly, we get `a_(n)=(4^(n)-1)+4^(n)x`
`or A=4^(n)-1 and B=4^(n)`
`or A+B+1=2^(2n+1),|A-B|=1, and `
`underset(n to oo)(lim)(4^(n)-1)/(4^(n))=underset(n to oo)(lim)(1-(1)/(4^(n)))=1`