Function f and g

Question

Let $f(x)$ mean that function $f$ ,applied to $x$,and $f^{n}(x)$ mean $f(f(........f(x)))$,that is $f$ applied to $x$ ,$n$ times.Let $g(x) = x+1$ and $h_{n}(x)=g^{n}(x).$Then what is $h_{9}^{8}(72)?$

Comments

First compute h₉(x)

Given: h_n(x)=gⁿ(x) , g(x)= x+1

g²(x)= g(g(x)) = g(x+1)=(x+1)+1 =x + 2*1

g³(x)= g(g(g(x))) = g(g²(x))=g(x+2)= (x+2)+1 = x+3 = x+ 3*1

Seeing this pattern we can infer that g⁹(x) = x+ 9*1=x +9

So, h₉(x)=g⁹(x)=x+9

Now we have to compute h₉⁸(x)  where x=72

h₉¹(x) = x+9

h₉²(x) = h₉(h₉(x)) = h₉(x+9) = (x+9) +9 = x+2*9

h₉³(x) = h₉(h₉(h₉(x))) = h₉(h₉²(x))= h₉(x+2*9) = (x+2*9) +9 = x+3*9

Again following the same pattern we can conclude that

h₉⁸(x) = x +8*9 =x+72

Put x=72 , it becomes 72+72=144

 

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Why dont you put that as an answer.

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@MiNiPanda

This is the nice method

I think this you add to the answer