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

Step-by-step solution

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 (f(x)-f ( x 2^n ) )= _ n (2 x (1- ( 1 2 )^ n+1 )-x ) & G(x)=x & _ r=1 ¹⁰ G (r^2 )= _ r=1 ¹⁰ r^2=385 aligned