15.2. Case with Repeated Characteristic Roots

Try these first

1. 1.

   Define what a characteristic root is.

   Hint

   Think about how you find sequences related to recurrence.

2. 2.

   What happens when the roots of a characteristic equation are repeated?

   Hint

   Consider the multiplicity of roots.

3. 3.

   What is the first step to solve a linear homogeneous recurrence equation?

   • Define initial conditions
   • Formulate the characteristic equation
   • Substitute values directly

   Hint

   Think about what helps in understanding sequence patterns.

4. 4.

   True or false: If characteristic roots are distinct, each root contributes one term to the general solution.

   • True
   • False

   Hint

   Consider how many times each root appears in forming solutions.

5. 5.

   Prove that the general form of the solution for a recurrence relation with one repeated root of multiplicity 3 must include polynomials of degree 2 and how would initial conditions apply?

   Hint

   Review the relationship between root multiplicities and polynomial degrees.

6. 6.

   If given a recurrence relation defined by $a_n = 2a_{n-1} - a_{n-2}$ with first terms as 5 and 6, derive the unique sequence.

   Hint

   Make sure to set up your equations correctly based on initial values.