BdMO 2020 Regional Higher Secondary P8

For students of class 11-12 (age 16+)
saifmd
Posts:15
Joined:Wed Feb 24, 2021 12:41 am
BdMO 2020 Regional Higher Secondary P8

Unread post by saifmd » Wed Mar 17, 2021 12:26 am

f f(x)=3x-2, then what is the value of x in the following equation? x=f(f(f(...f(x)...))) [2020 times]

User avatar
Anindya Biswas
Posts:264
Joined:Fri Oct 02, 2020 8:51 pm
Location:Magura, Bangladesh
Contact:

Re: BdMO 2020 Regional Higher Secondary P8

Unread post by Anindya Biswas » Wed Mar 17, 2021 1:20 am

saifmd wrote:
Wed Mar 17, 2021 12:26 am
$f\circ f(x)=3x-2$, then what is the value of $x$ in the following equation? $x=\underbrace{f(f(f(\dots f(x)\dots)))}_{2020}$
Let's define $g$ such that $g(x)=f(f(x))$. Let's denote $\underbrace{g(g(\dots g(x)\dots))}_{n}$ by $g^n(x)$.
We can apply induction to get $g^n(x)=3^nx-3^n+1$.
Given,
$g^{1010}(x)=3^{1010}x-3^{1010}+1=x\Rightarrow \boxed{x=1}$
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann

Post Reply