Let
$f(x)$ mean that the function $f$ is applied to $x$, and $f^n(x)$ mean $f(f(...f(x)))$, that is $f$ is applied to $x$, $n$ times. Let $g(x)=x+1$ and $h_n(x)=g^n(x)$. What is $h_n^m(0)$?
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What is
$g^n(x)$in terms of$x$and$n,$and how does that relate to the function$h$? -
How about trying to first find an expression
$h^2_n(x)$? -
By trying values of
$m,$give an expression for$g_n^m(x).$
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We have
$h_n(x)=g^n(x)=x+n$since$1$must be added$n$times. To compute$h_n^m(0)$we may first obtain$h_n^m(x),$and then evaluate it at$x=0.$Consider that$h_n^2(x)=h_n(h_n(x))=h_n(x+n)=x+2n.$We can hence inductively see that$h_n^m(x)=x+mn.$Therefore$h_n^m(0)=mn.$