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 $k=$
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.