Question:

Step 1: Understand the Recursive Structure of the Function We are given the initial function $f_1(x) = (x-2)^2$. The subsequent functions are defined recursively. From the pattern, $f_k(x)$ is defined such that $f_{k+1}(x) = (f_k(x) - 2)^2$ for $k \geq 1$. Each $f_k(x)$ is a polynomial, and we can express it in the form $f_k(x) = A_k + B_k x + C_k x^2 + D_k x^3 + \ldots$, where $A_k$ is the constant term and $B_k$ is the coefficient of $x$. Our goal is to find $B_8$. Step 2: Derive the Recurrence Relation for the Coefficient $B_k$ Let $f_k(x) = A_k + B_k x + C_k x^2 + \ldots$. Then, $f_{k+1}(x) = (f_k(x) - 2)^2$. Substitute the polynomial expansion of $f_k(x)$: $$f_{k+1}(x) = ((A_k + B_k x + C_k x^2 + \ldots) - 2)^2$$ $$f_{k+1}(x) = ((A_k - 2) + B_k x + C_k x^2 + \ldots)^2$$ Expanding this square, we look for the coefficient of $x$. The term $B_{k+1}x$ arises from the cross-product of the constant term and the $x$-term, i.e., $2 \cdot (A_k - 2) \cdot (B_k x)$. Therefore, the coefficient of $x$ in $f_{k+1}(x)$ is: $$B_{k+1} = 2(A_k - 2)B_k$$ Step 3: Calculate the Constant Term $A_k$ The constant term $A_k$ for any polynomial $f_k(x)$ is simply $f_k(0)$. For $f_1(x) = (x-2)^2$: $$f_1(x) = x^2 - 4x + 4$$ So, $A_1 = f_1(0) = 4$. For subsequent terms, using the recurrence $f_{k+1}(x) = (f_k(x) - 2)^2$: $$A_{k+1} = f_{k+1}(0) = (f_k(0) - 2)^2 = (A_k - 2)^2$$ Step 4: Determine the Sequence of $A_k$ Values Using the recurrence $A_{k+1} = (A_k - 2)^2$ with $A_1 = 4$: $$A_1 = 4$$ $$A_2 = (A_1 - 2)^2 = (4 - 2)^2 = 2^2 = 4$$ $$A_3 = (A_2 - 2)^2 = (4 - 2)^2 = 2^2 = 4$$ It is clear that $A_k = 4$ for all $k \geq 1$. This is a fixed point for the recurrence relation $A_{k+1} = (A_k - 2)^2$. Step 5: Determine the Sequence of $B_k$ Values Now we use the recurrence relation for $B_k$ derived in Step 2: $B_{k+1} = 2(A_k - 2)B_k$. Substitute the value of $A_k = 4$ for all $k \geq 1$: $$B_{k+1} = 2(4 - 2)B_k = 2(2)B_k = 4B_k$$ This is a simple geometric progression for $B_k$. First, we find $B_1$ from $f_1(x) = x^2 - 4x + 4$. $$B_1 = -4$$ Now, we can calculate the subsequent terms: $$B_2 = 4 \cdot B_1 = 4 \cdot (-4) = -16$$ $$B_3 = 4 \cdot B_2 = 4 \cdot (-16) = -64$$ $$B_4 = 4 \cdot B_3 = 4 \cdot (-64) = -256$$ $$B_5 = 4 \cdot B_4 = 4 \cdot (-256) = -1024$$ $$B_6 = 4 \cdot B_5 = 4 \cdot (-1024) = -4096$$ $$B_7 = 4 \cdot B_6 = 4 \cdot (-4096) = -16384$$ $$B_8 = 4 \cdot B_7 = 4 \cdot (-16384) = -65536$$ Alternatively, we can express $B_k$ in a general form: $B_k = B_1 \cdot 4^{k-1} = -4 \cdot 4^{k-1} = -4^k$. For $k=8$: $$B_8 = -4^8 = -(2^2)^8 = -2^{16} = -65536$$ Step 6: State the Final Answer The value of $B_8$ is $-65536$. Comparing this result with the given options: Option 1: -2048 Option 2: -32 Option 3: -1024 Option 4: -512 The calculated value $B_8 = -65536$ does not match any of the provided options. The final answer is $\boxed{-65536}$.