Find this value

Question

If f₀=1÷1-x and f_n(x)=f₀ (f_n-1(x)) where n=1,2,3,..............  then find f₂₀₁₄(2016)

Answer 1

f₀(x) = 1/(1-x)

 

f₁(x) = f₀(f₀(x)) = f₀(1/(1-x)) = 1/[1-(1/(1-x))]

 

                                            = 1/[-x/(1-x)] = (x-1)/x

 

f₂(x) = f₀(f₁(x)) = f₀[(x-1)/x]

 

                        = 1/[1-(x-1)/x] = 1/(1/x) = x

 

f₃(x) = f₀(f₂(x)) = f₀(x) = 1/(1-x)   etc

 

So, f₁ (x) = f₄(x) = f₇(x) = ... = f_3n-2 = (x-1)/x

 

2014 = 3(672)-2 

 

Therefore, f₂₀₁₄(x) = (x-1)/x

 

So, f₂₀₁₄(2016) = 2015/2016

 

Answer 2

f₀=1/(1-x)

f_n(x)=f₀ (f_n-1(x))

 

f_3n-2(x)=1-1/x, n=1,2,..

f_3n-1(x)=x, n=1,2,...

f_3n(x)=1/(1-x), n=1,2,..

 

2014=3*672-2. So

f₂₀₁₄(x)=f_3(672)-1(x)=1-1/x

f₂₀₁₄(2016)=1-1/2016=2015/2016