Question

If f(x + y) = f(x) + f(y) - xy - 1 for all $x,yϵR$ and f(1) = 1 then $f\left(n\right)=n,nϵN$ is true if?

• A. n = 1
• B. n = 1 and n = 2
• C. n is odd
• D. any value of n

Answer 1

The correct option is A n = 1

Since $f(x+y)=f\left(x\right)+f\left(y\right)-xy-1,\forall x,yϵR$
$\therefore f(x+1)=f\left(x\right)+f\left(1\right)-x-1[puttingy=1]$
$\Rightarrow f(x+1)=f\left(x\right)-x[\because f(1)=1]$
$\therefore f(n+1)=f\left(n\right)-n<f\left(n\right)\Rightarrow f(n+1)<f\left(n\right)$
So,
$f\left(n\right)<f(n-1)<f(n-2).....<f\left(3\right)<f\left(2\right)<f\left(1\right)$
$\therefore f\left(n\right)=n$ holds only for n = 1