Solve Recurrence Relation with Initial Conditions

Solve the recurrence relation $a_n = 3a_{n-1} - 2a_{n-2}$ with initial conditions $a_0 = 1$ and $a_1 = 3$.

Solving recurrence relations is a fundamental task in discrete mathematics, often encountered in algorithm analysis and combinatorics. Recurrence relations express each element of a sequence using the preceding elements, which can be integral in modeling problem states or sequential steps in computational problems. Understanding how to solve them is crucial for computer science students, especially for designing and analyzing algorithms.

For this problem, the relation is linear with constant coefficients, a common type of recurrence where methods like characteristic equations are applied. Initial conditions present a specific solution path, allowing us to determine particular constants for the relation's general solution. This problem introduces students to analyzing linear homogeneous recurrence relations, a foundational skill for more complex recurrence types like non-homogeneous or those involving variable coefficients.

After grasping basic recurrence structures and solutions, learners can expand this knowledge to derive closed-form solutions, facilitating more efficient computations than iterative methods. This understanding prepares you not only for theoretical applications but also for practical problem-solving scenarios often encountered in areas like dynamic programming, a key algorithm design paradigm in computer science.