Mathematics - Functions Question with Solution | TestHub

$\begin{array}{rl}& f(2x)-f(x)=x\\ & f(x)-f\left(\frac{x}{2}\right)=\frac{x}{2}\\ & \mathrm{f}\left(\frac{\mathrm{x}}{2}\right)-\mathrm{f}\left(\frac{\mathrm{x}}{4}\right)=\frac{\mathrm{x}}{4}\\ & \mathrm{f}\left(\frac{\mathrm{x}}{4}\right)-\mathrm{f}\left(\frac{\mathrm{x}}{8}\right)=\frac{\mathrm{x}}{8}\\ & f\left(\frac{x}{2^{n-1}}\right)-f\left(\frac{x}{2^n}\right)=\frac{x}{2^n}\\ & f(2x)-f\left(\frac{x}{2^n}\right)=x\left\{\frac{1-{\left(\frac{1}{2}\right)}^{n-1}}{1-\frac{1}{2}}\right\}\\ & f(x)-f\left(\frac{x}{2^n}\right)=2x\left(1-{\left(\frac{1}{2}\right)}^{n+1}\right)\\ & \mathrm{f}(\mathrm{x})+\mathrm{x}-\mathrm{f}\left(\frac{\mathrm{x}}{2^{\mathrm{n}}}\right)=2\mathrm{x}\left(1-{\left(\frac{1}{2}\right)}^{\mathrm{n}+1}\right)\\ & \underset{n\to \infty }{\lim }\left(f(x)-f\left(\frac{x}{2^n}\right)\right)=\underset{n\to \infty }{\lim }\left(2x\left(1-{\left(\frac{1}{2}\right)}^{n+1}\right)-x\right)\\ & G(x)=x\\ & \sum _{r=1}^{10}G\left(r^2\right)=\sum _{r=1}^{10}{r}^2=385\end{array}$