15. Solving Linear Homogeneous Recurrence Equations – Part II

Try these first

1. 1.

   Define a linear homogeneous recurrence equation.

   Hint

   Think about how sequences can be defined.

2. 2.

   What are characteristic roots?

   Hint

   Consider the roots of the equations from your algebra.

3. 3.

   What happens when roots of a characteristic equation are repeated?

   • Only one solution exists
   • The solution requires additional polynomial terms
   • No solution exists

   Hint

   Consider how sequences work with the same input.

4. 4.

   True or False? Initial conditions can determine the unique sequence from a recurrence relation.

   • True
   • False

   Hint

   Think about the role of conditions in forming a solution.

5. 5.

   Create a linear homogeneous recurrence relation of degree 4 with at least two repeated roots. Provide the characteristic polynomial and general solution.

   Hint

   Look for characteristic roots through polynomial factoring.

6. 6.

   Given that $a_n = 2a_{n-1} + a_{n-2}$ has initial conditions (0, 1), what is the specific sequence?

   Hint

   Substituting the initial conditions helps find constants in the general solution.