Question

Let $f$ be a function satisfying $f(x+y)=f\left(x\right)+f\left(y\right)$ for $x,yϵR$. If $f\left(1\right)=k$ then $f\left(n\right)$,$n\in N$, is equal to

• A. $k^n$
• B. $nk$
• C. $n^k$
• D. None of these

Answer 1

The correct option is B $nk$

Let $f\left(x\right)=\lambda \left(x\right)$ where $\lambda$ is a constant.

Hence

$f(x+y)$

$=\lambda (x+y)$

$=\lambda \left(x\right)+\lambda \left(y\right)$

$=f\left(x\right)+f\left(y\right)$

Now it has been given that $f\left(1\right)=k$

Therefore

$\lambda =k$

Hence

$f\left(x\right)=k\left(x\right)$

Therefore $f\left(n\right)$

$=k\left(n\right)$