14.9. Extension to Degree k Linear Homogeneous Recurrence Equations

Try these first

1. 1.

   What is the general form of a linear homogeneous recurrence equation?

   Hint

   Think about how each term relates to the previous terms.

2. 2.

   Define a characteristic equation.

   Hint

   Consider what replacing A_n with λ^n does.

3. 3.

   What does the characteristic equation reveal?

   • Roots of the sequence
   • Initial conditions
   • Terms of the sequence

   Hint

   Think about what you are solving with that equation.

4. 4.

   Is it possible to have multiple sequences satisfying the same recurrence relation?

   • True
   • False

   Hint

   Consider how initial conditions relate to the uniqueness of a solution.

5. 5.

   Given a recurrence relation A_n = 5A_{n-1} - 6A_{n-2}, derive its general solution and illustrate how the initial conditions would affect the constants.

   Hint

   Find roots, set general form, and plug in initial conditions.

6. 6.

   Analyze the recurrence given by A_n = A_{n-1} + 2A_{n-2} and identify if there are specific cases for constant definitions based on roots.

   Hint

   Solve the quadratic and understand how roots affect the sequence structure.