Second Order Linear Homogeneous Recurrence Relation with Initial Conditions

Solve the second-order linear homogeneous recurrence relation $a_n = 5a_{n-1} - 6a_{n-2}$ with initial conditions.

The problem presented requires solving a second-order linear homogeneous recurrence relation. Such recurrence relations are pivotal in discrete mathematics as they model various real-world phenomena including computational processes and sequences. At its core, solving these relations involves finding a general formula that allows you to compute any term given initial conditions.

For a second-order linear homogeneous recurrence relation, the general solution is typically expressed using characteristic equations or characteristic roots. This approach transforms the problem into finding the roots of a quadratic equation, which forms the basis of the solution. The nature of the roots—whether they are distinct, repeated, or complex—determines the form of the solutions. When the roots are distinct real numbers, the general solution is a combination of exponential terms.

When roots are repeated, additional factors involving the term’s position come into play. If the roots are complex, the terms incorporate trigonometric functions, adding another layer of complexity to the solution, though this is less common in basic problems. In this problem, recognizing the pattern and structure of the given recurrence relation is crucial. Attention to the rules of manipulating sequences, understanding initial conditions, and correctly applying formulae derived from characteristic roots is vital. This problem also serves as a great exercise in reinforcing algebraic manipulation skills and abstract reasoning in series and sequences often encountered in discrete mathematics.