I am not sure how to tackle the following question mentioned below:
Let $h(x) = 2x$
Let $a(x) = x+3$
Compute the value of $f(5)$ where $f(x)$ is defined as $f(x)= h ∘ a ∘ h ∘ a$ $ ∘ $ ... $ ∘ $ $ h ∘ a(x)$, when ($h ∘ a$) appears $2020$ times in $f(x)$.
I thought about using the fact that $2^{2020} + 2^{2019} + ... + 2^2 + 2 = 2^{2021}-2$, but I'm not sure I am doing it correctly.
What I attempted doing is, simply plugging $2^{2021}-2$ into $a(x)$, which gave me $((2^{2021}-2)x +3)$, so I plugged $x = 5$ in and got $((2^{2021}-2)*5+3)$.
Is that the right way?
$$\begin{align}h\circ a&=2(x+3)\\h\circ a\circ h\circ a&=2(2(x+3)+3)=2^2x+2^2\cdot 3+2\cdot 3\\h\circ a \circ h\circ a\circ h\circ a &=2(2^2x+2^2\cdot3+2\cdot3+3)=2^3x+2^3\cdot3+2^2\cdot3+2\cdot3\end{align}$$ If you apply $2020$ times you get $$f(x)=2^{2020}x+(2^{2020}+...+2^3+2^2+2)3$$ Now you can apply your formula for the sum $$f(x)=2^{2020}x+6(2^{2020}-1)$$