Track / Stage 5 / 81 of 400 #681 of 2000

Problem 681

AIME late

Algebra Difficulty 5.2 Find the answer

Example 6 Try to find $f^{(n)}(x)$ using the fixed point method, where $f(x)=a x+b$ .

Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.

However, it seems there is a misunderstanding in your request. The text you provided is already in English. If you meant to translate the problem statement into English, here it is:

Example 6 Try to use the fixed point method to find $f^{(n)}(x)$ , where $f(x)=a x+b$ .

Your answer

Official solution

Solve: Let $f(x)=x$ , we get the unique fixed point of $f(x)$ as $x_{0}=\frac{b}{1-a}$ . Then $f(x)=a\left(x-x_{0}\right)+f\left(x_{0}\right)$ .
$\begin{aligned} f^{(2)}(x) & =a f(x)+b=a^{2}\left(x-x_{0}\right)+a f\left(x_{0}\right)+b \\ & =a^{2}\left(x-x_{0}\right)+a x_{0}+b=a^{2}\left(x-x_{0}\right)+f\left(x_{0}\right), \cdots \\ f^{(n)}(x) & =a^{\prime \prime}\left(x-x_{0}\right)+f\left(x_{01}\right)=a^{n}\left(x-\frac{b}{1-a}\right)+\frac{b}{1-a} \end{aligned}$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.