<s>[INST] What are the coefficients of the polynomial inductively defined as $f_1=(x-2)^2\,\,\,;\,\,\,f_{n+1}=(f_n-2)^2$? 
Let $\{f_n(x)\}_{n\in \Bbb N}$ be a sequence of polynomials defined inductively as
  $$\begin{matrix} f_1(x) & = & (x-2)^2 &
\\ f_{n+1}(x)& = & (f_n(x)-2)^2, &\text{ for all }n\ge 1.\end{matrix}$$
Let $a_n$ and $b_n$ respectively denote the constant term and the coefficient of $x$ in $f_n(x)$. Then
  
  
*
  
*$(\text A)\,\, a_n=4, b_n=-4^n$
  
*$(\text B)\,\,a_n=4, b_n=-4n^2$
  
*$(\text C)\,\, a_n=4^{(n-1)!}, b_n=-4^n$
  
*$(\text C)\,\, a_n=4^{(n-1)!}, b_n=-4n^2$
  

Please help me how to solve this problem. I am not able to solve it.
The problem can be found on this link.
 [/INST] HINT:
$f_1(x)=(x-2)^2=4-4x+x^2$
$f_2(x)=(f_1(x)-2)^2$
$=(x^2-4x+4-2)^2=(2-4x+x^2)^2=2^2+2\cdot2(-4x)+\cdots=4-4^2x+\cdots$
$f_3(x)=(f_2(x)-2)^2$
$=(4-2^4x+\cdots-2)^2=(2-2^4x+\cdots)^2=2^2-2\cdot2\cdot4^2x+\cdots=2^2-4^3x+\cdots$
$f_4(x)=(f_3(x)-2)^2$
$=(2^2-4^3x+\cdots-2)^2=(2-4^3x+\cdots)^2=2^2-2\cdot2\cdot2^6x+\cdots=2^2-4^4x+\cdots$
$f_5(x)=(f_4(x)-2)^2$
$=(4-4^4x+\cdots)^2=(2-4^4x+\cdots)^2=2^2-2\cdot2\cdot2^8x+\cdots=2^2-4^5x+\cdots$
So, $a_n=4$
$b_n=-4^{(n \text{ th term of } 1,2,3,4,5)}=-4^n$
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