Question

8. $C_k$ is equal to

• A. $\frac{4^{2k}-4^k}{3}$
• B. $4^{2k-2}$
• C. $\frac{4^{2k}+4^k}{5}$
• D. $4^{k+1}$

Answer 1

Answer: (A)

$\frac{4^{2k}-4^k}{3}$

Answer 2

The question asks for the value of $C_k$ and provides four options. However, the definition of $C_k$ is missing from the question statement. Without a definition, it is impossible to determine the correct expression for $C_k$.

In mathematical problems, $C_k$ typically represents a term in a sequence, a combination, or a solution to a recurrence relation. The given options are expressions involving powers of 4.

Let's analyze the options: (A) $C_k = \frac{4^{2k}-4^k}{3}$ (B) $C_k = 4^{2k-2}$ (C) $C_k = \frac{4^{2k}+4^k}{5}$ (D) $C_k = 4^{k+1}$

We can test small integer values for $k$ to see if any option yields non-integer values, which might make it less likely if $C_k$ is expected to be an integer (e.g., counting problem).

For $k=1$: (A) $C_1 = \frac{4^{2(1)}-4^1}{3} = \frac{16-4}{3} = \frac{12}{3} = 4$ (B) $C_1 = 4^{2(1)-2} = 4^0 = 1$ (C) $C_1 = \frac{4^{2(1)}+4^1}{5} = \frac{16+4}{5} = \frac{20}{5} = 4$ (D) $C_1 = 4^{1+1} = 4^2 = 16$

For $k=2$: (A) $C_2 = \frac{4^{2(2)}-4^2}{3} = \frac{4^4-4^2}{3} = \frac{256-16}{3} = \frac{240}{3} = 80$ (B) $C_2 = 4^{2(2)-2} = 4^{4-2} = 4^2 = 16$ (C) $C_2 = \frac{4^{2(2)}+4^2}{5} = \frac{4^4+4^2}{5} = \frac{256+16}{5} = \frac{272}{5} = 54.4$ (D) $C_2 = 4^{2+1} = 4^3 = 64$

Option (C) yields a non-integer value for $k=2$. In most contexts where $C_k$ represents a count or a term in a sequence, it is expected to be an integer. This makes option (C) less likely.

Options (A) and (C) are of the form $A \cdot (4^2)^k + B \cdot 4^k$, implying they could be solutions to a linear homogeneous recurrence relation with constant coefficients. Let $C_k = A \cdot (16)^k + B \cdot (4)^k$. The characteristic equation would have roots $16$ and $4$. So, $x^2 - (16+4)x + (16 \times 4) = 0 \implies x^2 - 20x + 64 = 0$. The recurrence relation is $C_k = 20 C_{k-1} - 64 C_{k-2}$.

Let's check if option (A) satisfies this recurrence: $C_k = \frac{4^{2k}-4^k}{3}$ $20 C_{k-1} - 64 C_{k-2} = 20 \left(\frac{4^{2(k-1)}-4^{k-1}}{3}\right) - 64 \left(\frac{4^{2(k-2)}-4^{k-2}}{3}\right)$ $= \frac{1}{3} \left[ 20 \cdot 4^{2k-2} - 20 \cdot 4^{k-1} - 64 \cdot 4^{2k-4} + 64 \cdot 4^{k-2} \right]$ $= \frac{1}{3} \left[ 20 \cdot \frac{4^{2k}}{16} - 20 \cdot \frac{4^k}{4} - 64 \cdot \frac{4^{2k}}{256} + 64 \cdot \frac{4^k}{16} \right]$ $= \frac{1}{3} \left[ \frac{5}{4} \cdot 4^{2k} - 5 \cdot 4^k - \frac{1}{4} \cdot 4^{2k} + 4 \cdot 4^k \right]$ $= \frac{1}{3} \left[ \left(\frac{5}{4} - \frac{1}{4}\right) 4^{2k} + (-5+4) 4^k \right]$ $= \frac{1}{3} \left[ \frac{4}{4} 4^{2k} - 1 \cdot 4^k \right]$ $= \frac{4^{2k}-4^k}{3} = C_k$. So, option (A) is a valid solution to the recurrence $C_k = 20 C_{k-1} - 64 C_{k-2}$ with initial conditions $C_0=0$ and $C_1=4$.

Without the definition of $C_k$, the question is incomplete and cannot be definitively answered. However, if this question is part of a larger problem set where $C_k$ was defined earlier, or if it refers to a very specific, commonly known sequence not explicitly stated, then one of these options would be correct. Given the incomplete nature, it's impossible to select a unique answer based solely on the provided information.

If forced to choose the most plausible option based on typical exam question patterns (integer values, common recurrence relation solutions), option (A) is a strong candidate because it produces integers for small $k$ and is a solution to a standard linear recurrence relation. Option (C) is less likely due to non-integer values. Options (B) and (D) are simpler exponential forms, but without context, there's no reason to prefer them.

Conclusion: The question is incomplete as the definition of $C_k$ is missing. Therefore, a definitive answer cannot be provided.