JEE Math Practice Question

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function satisfying $f(0)=1$ and $f(2 x)-f(x)=x$ for all $x \in \mathbb{R}$. If $\lim \limits_{n \rightarrow \infty}\left\{f(x)-f\left(\frac{x}{2^n}\right)\right\}=G(x)$, then $\sum\limits_{r=1}^{10} G\left(r^2\right)$ is equal to

1. A.215
2. B.420
3. C.385
4. D.540