Let's break down and solve the two parts of this problem.

Part (a):

We are given the recurrence relation: $a_{n+1} - a_n = 3n, \quad n \geq 0, \quad a_0 = 1.$ We are tasked with finding the generating function of $a_n$.

Step 1: Solve the recurrence relation

We can rewrite the recurrence as: $a_{n+1} = a_n + 3n.$ To solve this, let's first find the explicit form of $a_n$.

The general solution can be written as: $a_n = a_0 + \sum_{k=0}^{n-1} 3k.$ The sum of the first $n$ integers is: $\sum_{k=0}^{n-1} k = \frac{(n-1)n}{2}.$ Thus, we have: $a_n = 1 + 3 \cdot \frac{(n-1)n}{2} = 1 + \frac{3n(n-1)}{2}.$ Simplifying the expression: $a_n = 1 + \frac{3n^2 - 3n}{2}.$

Step 2: Generating function

The generating function $A(x)$ for the sequence $a_n$ is defined as: $A(x) = \sum_{n=0}^{\infty} a_n x^n.$ We substitute $a_n = 1 + \frac{3n^2 - 3n}{2}$ and compute the generating function.

Part (b):

We are given the recurrence relation: $a_{n+2} - 11a_{n+1} + 24a_n = 0, \quad a_0 = 1, \quad a_1 = 2.$ This is a second-order linear recurrence relation with constant coefficients.

Step 1: Characteristic equation

The characteristic equation corresponding to this recurrence is: $r^2 - 11r + 24 = 0.$ Solving this quadratic equation: $r = \frac{11 \pm \sqrt{11^2 - 4 \cdot 1 \cdot 24}}{2 \cdot 1} = \frac{11 \pm \sqrt{121 - 96}}{2} = \frac{11 \pm \sqrt{25}}{2}.$ Thus, the roots are: $r = \frac{11 \pm 5}{2}.$ The two roots are $r = 8$ and $r = 3$.

Step 2: General solution

The general solution to the recurrence relation is: $a_n = c_1 \cdot 8^n + c_2 \cdot 3^n.$ Using the initial conditions $a_0 = 1$ and $a_1 = 2$, we can solve for $c_1$ and $c_2$.

1. For $n = 0$: $a_0 = c_1 \cdot 8^0 + c_2 \cdot 3^0 = c_1 + c_2 = 1.$

2. For $n = 1$: $a_1 = c_1 \cdot 8^1 + c_2 \cdot 3^1 = 8c_1 + 3c_2 = 2.$

We now solve the system of equations: $c_1 + c_2 = 1,$ $8c_1 + 3c_2 = 2.$ From the first equation, $c_2 = 1 - c_1$. Substituting into the second equation: $8c_1 + 3(1 - c_1) = 2,$ $8c_1 + 3 - 3c_1 = 2,$ $5c_1 = -1,$ $c_1 = -\frac{1}{5}.$ Thus, $c_2 = 1 - \left(-\frac{1}{5}\right) = \frac{6}{5}$.

Final solution:

The solution to the recurrence is: $a_n = -\frac{1}{5} \cdot 8^n + \frac{6}{5} \cdot 3^n.$

Would you like further clarification or a deeper breakdown of either part? Here are five related questions:

1. How can we interpret the generating function of a sequence?
2. Can we use matrix methods to solve recurrence relations?
3. How would the recurrence relation change if different initial conditions were provided?
4. What is the significance of the characteristic equation in solving linear recurrence relations?
5. How does one handle non-homogeneous recurrence relations?

Tip: Solving a linear recurrence relation often involves finding the roots of the characteristic equation, which gives insight into the general behavior of the sequence.