15.6. General Case for Degree k with Repeated Roots

Try these first

1. 1.

   What is a homogeneous recurrence equation?

   Hint

   Think about how each term relates to those before it.

2. 2.

   What is meant by characteristic roots?

   Hint

   How do roots influence the structure of a polynomial?

3. 3.

   What is the main consequence of having repeated roots in a characteristic equation?

   • Each root contributes equally to the sequence
   • The general form of the solution includes polynomials
   • Repeated roots have no effect on the solution

   Hint

   Think about the structure of the general solution.

4. 4.

   True or False: The sum of multiplicities of roots must equal the degree of the polynomial.

   • True
   • False

   Hint

   Recall how roots are defined in terms of their counts.

5. 5.

   Given the recurrence relation a_n = 4a_{n-1} - 4a_{n-2}, with initial conditions a_0 = 2, a_1 = 6, derive the unique sequence.

   Hint

   Think about how repeated roots modify coefficients and constants in the general solution.

6. 6.

   Solve the recurrence relation a_n + a_{n-1} + a_{n-2} = 0 and find its unique sequence given initial conditions 1, -1, -1.

   Hint

   Identify the multiplicity of the roots to establish the correct polynomial degrees.