Functions
Continuous functional equation
nta_pyq_2025_apr
Grade 12
Question:
Let f : R$\to$R be a continuous function satisfying f$(0) = 1$and f$(2x) - f$$(x) = x$for all x$\ in $R. If , then$\sum$is equal to x 10 2 limn$\to$$\infty${f$(x) - f$($)} = G(x)$G (r ) n$r=1$2
Step-by-Step Solution
Key Concept: Write the given equation with the transformed input and solve the resulting simultaneous equations for the function value.
f$(2x) - f$$(x) = x$x x (2) f$(x) - f$($) = 2$2 x x x f ($) - f$($) = 2$4 4 x x x f ($) - f$($) = 4$8 8 x x x f ($) - f$($) = n-1$n n 2 2 2$n-1${ 1 ⎫ ⎪ ⎪$1 - ($) ⎪ ⎪ x 2 f$(2x) - f$($) = x$⎬ n 1 2 ⎪ 1 - ⎪ ⎪ 2 ⎭ ⎪$n+1$x 1 f$(x) - f$($) = 2x$$(1 - ($) ) n 2 2$n+1$x 1 f$(x) + x - f$($) = 2x$$(1 - ($) ) n 2 2$n+1$x 1 lim (f$(x) - f$($)) = lim$(2x$(1 - ($)$) - x)$n n$\to$$\infty$2 n$\to$$\infty$2$G(x) = x$10 10 2 2$\sum$G (r ) =$\sum$$r = 385$$r=1$$r=1$
Correct Answer: 2