Let f(x) be a real differentiable function such that f(0)=1 and f(x+y)=f(x) f^ (y)+f^ (x) f(y) for all x, y R . Then _ n =1 ¹⁰⁰ _ e f( n ) is equal to :

Step-by-step solution

f(x+y)=f(x) f(y)+f(x) f(y), x, y R ....(i) And f(0)=1 ....(ii) Now replace x by zero and y by zero we get aligned & f(0)=f(0) f(0)+f(0) f(0) & 1=f(0)+f(0) & f^ (0)= 1 2 ...(iii) aligned Now replace y by zero in equation (i), we get f(x)= 1 2 f(x)+f^ (x) or, 1 2 f(x)=f^ (x) then f^ (x) f(x) = 1 2 hence |f(x)|= x 2 +c Put x=0 , we get c=0 |f(x)|= x 2 Then _ n=1 ¹⁰⁰ (f( ))= ( 1 2 + 2 2 + 3 2 + + 100 2 )= 5050 2 =2525