Question

State whether the given statement is true or false

If f(x) satisfies the relation, f(x+y)=f(x)+f(y) for all $x,yϵR$ and f(1)=5, then find ${\sum }_{n=1}^{m}f\left(n\right)$.

• A. True
• B. False

Answer 1

The correct option is B False

We have, $f(x+y)=f\left(x\right)+f\left(y\right)$
Put $x=y=0$ we get, $f\left(0\right)=0\Rightarrow$ (1)
Now,
$f^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{f(x+h)-f\left(x\right)}{h}=\underset{h\to 0}{lim}\frac{f\left(x\right)+f\left(h\right)-f\left(x\right)}{h}$
$f^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{f\left(h\right)-f\left(0\right)}{h}=f^{\prime }\left(0\right)$, using (1)
Integrating we get, $f\left(x\right)=f^{\prime }\left(0\right)\cdot x+c$
Also $f\left(0\right)=0\Rightarrow c=0$
It is given that, $f\left(1\right)=5\Rightarrow f^{\prime }\left(0\right)=5$
Hence $f\left(x\right)=5x$
$\therefore \sum _{n=1}^{m}f\left(n\right)=5\times \sum _{n=1}^m\left(n\right)=\frac{5}{2}m(m+1)=$ odd number