Question

Let $f: R \rightarrow R$ be a function which satisfies $f ( x + y )= f ( x )+ f ( y ) \forall x , y \in R .$ If $f (1)=2$ and $g(n)=\displaystyle\sum_{k=1}^{(n-1)} f(k), n \in N$ then the value of $n,$ for which $g(n)=20,$ is :

• A. 5
• B. 9
• C. 20
• D. 4

Answer 1

Answer: (A)

$f(x+y)=f(x)+f(y)$
$\Rightarrow f(n)=n f(1)$
$f(n)=2 n$
$g(n)=\displaystyle\sum_{k=1}^{n-1} 2 n=2\left(\frac{(n-1) n}{2}\right)=n(n-1)$
$g(n)=20 \Rightarrow n(n-1)=20$
$n=5$