Let-f-0-x-1-1-x-and-f-n-x-f-0-f-n-1-x-n-1-2-3-Evaluate-f-2018-2018-

Question Number 111149 by Aina Samuel Temidayo last updated on 02/Sep/20

Let f_{0} (x) = \frac{1}{1 - x} and f_{n} (x)

= f_{0} (f_{n - 1} (x)), n = 1, 2, 3, \dots Evaluate

f_{2018} (2018)

Commented by kaivan.ahmadi last updated on 02/Sep/20

f_{1} (x) = f_{0} (f_{0} (x)) = f_{0} (\frac{1}{1 - x}) = \frac{1}{1 - \frac{1}{1 - x}} = \frac{1}{\frac{1 - x - 1}{1 - x}} = \frac{1 - x}{- x} = 1 - \frac{1}{x}

f_{2} (x) = \frac{1}{1 - 1 + \frac{1}{x}} = x

f_{3} (x) = \frac{1}{1 - x} = f_{0} (x)

\Rightarrow f_{n} (x) = f_{i}; i = 0 o r 1 o r 2 a n d n \overset{3}{\equiv} i

2018 \overset{3}{\equiv} 2 \Rightarrow f_{2018} (x) = f_{2} (x) = x

\Rightarrow f_{2018} (2018) = 2018

Commented by Aina Samuel Temidayo last updated on 03/Sep/20

Please I {don}^{'} t understand these lines .

I will be glad if you can help shed more

light on it .

\Rightarrow f_{n} (x) = f_{i}, i = 0 or 1 or 2 and n \overset{3}{\equiv} i

2018 \overset{3}{\equiv} 2 \Rightarrow f_{2018} (x) = f_{2} (x) = x

Commented by kaivan.ahmadi last updated on 03/Sep/20

f_{0} (x) = \frac{1}{1 - x}

f_{1} (x) = 1 - \frac{1}{x}

f_{2} (x) = x

\Rightarrow f_{3} (x) = f_{0} (x) f_{5} (x) = f_{1} (x) f_{6} (x) = f_{2} (x)

s o f_{n} (x) i s a o n e o f t h e f_{0}, f_{1}, f_{2}

f_{n} (x) = {\begin{cases} f_{0} (x); n = 3 k \\ f_{1} (x); n = 3 k + 1 (k \in Z) \\ f_{2} (x); n = 3 k + 2 \end{cases}

n o w s i n c e 2018 = 3 (672) + 2 w e h a v e

f_{2018} (x) = f_{2} (x) = x

Commented by Aina Samuel Temidayo last updated on 03/Sep/20

I understand now . Thanks .