Question

Let c , k∈ R. If f(x) = (c + 1)x 2 + (1 – c 2)x + 2 k and f(x + y) = f(x) + f(y) – xy , for all x , y ∈ R, then the value of |2(f(1) + f(2) + f(3) + ......+f (20))| is equal to _____.

Answer 1

The correct answer is: 3395

f(x) is polynomial

Put y = 1/x in given functional equation we get

$f(x+\frac{1}{x})=f(x)+f(\frac{1}{x})-1$

⇒ 2(c + 1) = 2 K – 1 …(1)

and put x = y = 0 we get

$f(0)=2+f(0)-0⇒f(0)=0⇒k=0$

∴ k = 0 and 2 c = –3 ⇒ c = $–\frac{3}{2}$

by simplifying we will get

$=|-\frac{6790}{2}|=3395$