Let $f$ be a real-valued function such that $f(x+y)=f(x)+f(y) \quad \forall x, y \in$ $\boldsymbol{R} .$ Define a function $\phi$ by $\phi(x)=c+f(x), x \in \boldsymbol{R},$ where $c$ is a real constant. Show that for every positive integer $n$ $$ \phi^{n}(x)=\left(c+f(c)+f^{2}(c)+\ldots . .+f^{n-1}(c)\right)+f^{n}(x) $$ where, for a real-valued function $g, \quad g^{n}(x)$ is defined by $$ g^{0}(x)=0, g^{1}(x)=g(x), g^{k+1}(x)=g\left(g^{k}(x)\right) $$
I have tried using Induction, but it seems lengthy.Anyother approach will greatly be appreciated!
Induction step: $$\begin{align*}\phi^{n+1}(x)&=\phi(\phi^n(x))\\&=c+f(\phi^n(x))\\ &=c+f(c+f(c)+f^2(c)+\cdots+f^{n-1}(c)+f^{n}(x))\\ &=c+f(c)+f^2(c)+f^3(c)+\cdots+f^n(c)+f^{n+1}(x).\end{align*}$$