Let f:R R be a continuous function satisfying f(0)=1 and f(2x)-f(x)=x for all x R.

Question

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

Accepted Answer

$\begin{gathered} f(0)=1,f(2x)-f(x)=x \ x	o \frac{x}{2} \ f(x)-f(\frac{x}{2})=\frac{x}{2}\, \end{gathered}$

Again, Replace $x	o \frac{x}{2}$

$f(\frac{x}{2})-f(\frac{x}{{2}^{2}})=\frac{x}{{2}^{2}}\,$

: $\,$ : $\,$ :

$\begin{gathered} f(\frac{x}{{2}^{n-1}})-f(\frac{x}{{2}^{n}})=\frac{x}{{2}^{n}} \ (1)+(2)+(3)+\ldots +(n) \ f(x)-f(\frac{x}{{2}^{n}})=\frac{x}{2}+\frac{x}{{2}^{2}}+\ldots .+\frac{x}{{2}^{n}} \ \underset{n	o \infty}{lim}(f(x)-f(\frac{x}{{2}^{n}}))=\underset{n	o \infty}{lim}(\frac{x}{2}+\frac{x}{{2}^{n}}+\ldots +\frac{x}{{2}^{n}}) \end{gathered}$

$\begin{gathered} f(x)-f(0)=\frac{\frac{x}{2}}{\frac{1}{2}} \ \Rightarrow G(x)=x \ \Rightarrow G({r}^{2})={r}^{2} \ \Rightarrow \sum _{r=1}^{10}G({r}^{2})=\sum _{r=1}^{10}G({r}^{2}) \ =\frac{(10)(11)(21)}{6}=(55)7 \ \Rightarrow 385 \end{gathered}$

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