Track / Stage 4 / 335 of 340 #595 of 1964

Problem 595

AMC 12 late, AIME early

Algebra Difficulty 5.0 Find the answer

4. Let $f(x)=a x+b$ (where $a, b$ are real numbers), $f_{1}(x)=f(x), f_{n+1}(x)=f\left(f_{n}(x)\right), n=1$ , $2,3, \cdots$ , If $2 a+b=-2$ , and $f_{k}(x)=-243 x+244$ , then $k=$

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Official solution

4. 5 Detailed Explanation: We can first find $f_{1}(x), f_{2}(x), f_{3}(x)$ , and by induction we get $f_{k}(x)=a^{k} x+\frac{1-a^{k}}{1-a} \cdot b$ . Substituting the known values, we can solve for the result.

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