Question

If $f:R\to R$ satisfies $f(x+y)=f\left(x\right)+f\left(y\right)$, for all $x,y\in R$ and $f\left(1\right)=7$ ,then ${\sum }_{r=1}^{n}f\left(r\right)$ is

• A. $\frac{7n(n+1)}{2}$
• B. $\frac{7n}{2}$
  
• C. $\frac{7(n+1)}{2}$
  
• D. 7n+(n+1)

Answer 1

The correct option is A

$\frac{7n(n+1)}{2}$

$f(x+y)=f\left(x\right)+f\left(y\right)\text{Function should be}f\left(x\right)=mxF\left(1\right)=7;\therefore m=7,f\left(x\right)=7x{\sum }_{r=1}^{n}f\left(r\right)=7{\sum }_1^{n}r=\frac{7n(n+1)}{2}$