Let's solve the problem step by step as requested. **Given:** \[ y_k = A + (-1)^k B + Ck \] Our goal: **Derive a difference equation by eliminating the constants \(A\), \(B\), and \(C\).** --- ### **Step 1: Write down the expression for \(y_{k+1}\) and \(y_{k+2}\)** **Explanation:** To form a difference equation, we need expressions for consecutive terms. \[ \begin{align*} y_k &= A + (-1)^k B + Ck \\ y_{k+1} &= A + (-1)^{k+1}B + C(k+1) \\ &= A - (-1)^k B + Ck + C \\ y_{k+2} &= A + (-1)^{k+2}B + C(k+2) \\ &= A + (-1)^k B + Ck + 2C \\ \end{align*} \] --- ### **Step 2: Form a linear combination to eliminate constants** **Explanation:** We want to eliminate \(A\), \(B\), and \(C\). Let’s look for a linear combination of \(y_{k+2}\), \(y_{k+1}\), and \(y_k\) that eliminates the constants. Let’s consider: \[ y_{k+2} - 2y_{k+1} + y_k \] Now, substitute the expressions: \[ \begin{align*} &[A + (-1)^k B + Ck + 2C] \\ -&2[A - (-1)^k B + Ck + C] \\ +&[A + (-1)^k B + Ck] \\ \end{align*} \] Let’s expand: - \(A + (-1)^k B + Ck + 2C\) - \(-2A + 2(-1)^k B - 2Ck - 2C\) - \(+A + (-1)^k B + Ck\) Combine: - \(A - 2A + A = 0\) - \((-1)^k B + 2(-1)^k B + (-1)^k B = 4(-1)^k B\) - \(Ck - 2Ck + Ck = 0\) - \(2C - 2C = 0\) So the result is: \[ y_{k+2} - 2y_{k+1} + y_k = 4(-1)^k B \] --- ### **Step 3: Further eliminate the \(B\) term** **Explanation:** We see the right side is still \(4(-1)^k B\). We need another step to eliminate \(B\). Apply the same operation to the next value, i.e., for \(k+1\): \[ y_{k+3} - 2y_{k+2} + y_{k+1} = 4(-1)^{k+1} B = -4(-1)^k B \] Now, add the two equations: \[ [y_{k+2} - 2y_{k+1} + y_k] + [y_{k+3} - 2y_{k+2} + y_{k+1}] = 4(-1)^k B - 4(-1)^k B = 0 \] So, \[ y_{k+3} - y_{k+2} - y_{k+1} + y_k = 0 \] --- ## **Final Answer** The required difference equation is: \[ \boxed{y_{k+3} - y_{k+2} - y_{k+1} + y_k = 0} \] --- ### **Summary of Steps** 1. **Write expressions for \(y_{k}\), \(y_{k+1}\), \(y_{k+2}\):** Set up the forms needed to create a difference equation. 2. **Form a difference equation involving those terms:** Combine \(y_{k+2} - 2y_{k+1} + y_k\) to eliminate constants \(A\) and \(C\), leaving a term with \(B\). 3. **Eliminate the remaining constant (\(B\)):** Use the same operation at the next index and add the equations, resulting in a difference equation with all constants eliminated. **Thus, the final difference equation is:** \[ \boxed{y_{k+3} - y_{k+2} - y_{k+1} + y_k = 0} \]