### 4. Given recurrence relation \[ a_n = -a_{n-1} + n \] with initial condition \[ a_0 = 2 \] --- ### (a) Associated homogeneous recurrence relation **Solution:** \[ a_n = -a_{n-1} \] --- ### (b) General solution to the homogeneous recurrence \((a_n^{(h)})\) **Solution:** The characteristic equation has a single root \(-1\), so the general solution is \[ a_n^{(h)} = \alpha(-1)^n \] --- ### (c) Particular solution \((a_n^{(p)})\) **Solution:** Since the non-homogeneous part is \(n\), assume a particular solution of the form \[ a_n^{(p)} = cn + d \] Substitute into the recurrence relation: \[ cn + d = -\big(c(n-1) + d\big) + n \] \[ = -cn + c - d + n \] Rearranging: \[ 2cn - c + 2d - n = 0 \] \[ n(2c - 1) + (2d - c) = 0 \] Set each coefficient equal to zero: \[ 2c - 1 = 0 \Rightarrow c = \frac{1}{2} \] \[ 2d - \frac{1}{2} = 0 \Rightarrow d = \frac{1}{4} \] Therefore, the particular solution is \[ a_n^{(p)} = \frac{1}{2}n + \frac{1}{4} \]