Let f: ℝ arrow ℝ be a continuous function satisfying f(0)=1 and f(2 x)-f(x)=x for all x ∈ ℝ . If _ n arrow ∞ \ f(x)-f ((x)/(2^n))\ =G(x) , then ∑_ r=1 ^ 10 G(r^2) is equal to

Correct answer: B. 385 aligned & f(2 x)-f(x)=x & f(x)-f((x)/(2))=(x)/(2) & f( x 2 )-f( x 4 )= x 4 & f( x 4 )-f( x 8 )= x 8 & f( x 2^ n-1 )-f((x)/(2^n))=(x)/(2^n) & f(2 x)-f((x)/(2^n))=x\ 1-((1)/(2))^ n-1 1-(1)/(2) \ & f(x)-f((x)/(2^n))=2 x(1-((1)/(2))^ n+1 ) & f(x)+x-f( x 2^ n )=2 x(1-((1)/(2))^ n+1 ) & _ n arrow ∞ (f(x)-f((x)/(2^n)))= _ n arrow ∞ (2 x(1-((1)/(2))^ n+1 )-x) & G(x)=x & ∑_ r=1 ^ 10 G(r^2)=∑_ r=1 ^ 10 r^2=385 aligned

JEE Main 20254 AprilMorning Shift

Exam: JEE Main
Year: 2025
Date: 4 April
Shift: Morning Shift
Subject: Mathematics
Chapter: Functions

JEE Main 2025 · 4 April · Morning Shift · Mathematics · Functions

Let f: ℝ arrow ℝ be a continuous function satisfying f(0)=1 and f(2...

Let

f: \mathbb{R} \rightarrow \mathbb{R}

be a continuous function satisfying

f(0)=1

and

f(2 \mathrm{x})-f(\mathrm{x})=\mathrm{x}

for all

\mathrm{x} \in \mathbb{R}

. If

\lim _{n \rightarrow \infty}

\left\{f(x)-f

\left(\frac{x}{2^n}\right)\right\}=G(x)

, then

\sum_{r=1}^{10} G\left(r^2\right)

is equal to

Options

A540
B385
C420
D215

Correct answer

B. 385

Step-by-step solution

\begin{aligned}<br/>& f(2 x)-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(2 x)-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)=2 x\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) \\ & \lim _{n \rightarrow \infty}\left(f(x)-f\left(\frac{x}{2^n}\right)\right)=\lim _{n \rightarrow \infty}\left(2 x\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<br/>\end{aligned}

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