15.3. General Form and Initial Conditions

Try these first

1. 1.

   What is a linear homogeneous recurrence equation?

   Hint

   Think about how these equations relate sequences.

2. 2.

   What do characteristic roots represent?

   Hint

   What do we derive from the characteristic equation?

3. 3.

   What do we call the values derived from the characteristic polynomial of a recurrence relation?

   • Initial Conditions
   • General Solutions
   • Characteristic Roots

   Hint

   Think about what those values represent.

4. 4.

   True or False: Initial conditions are not necessary for determining unique solutions in recurrence equations.

   • True
   • False

   Hint

   Consider multiple sequences that can satisfy a recurrence.

5. 5.

   Consider the recurrence relation a_n = 2a_{n-1} - a_{n-2}. Find and characterize its general solution.

   Hint

   Identify the polynomial from the given recursion.

6. 6.

   Given the initial conditions a_0 = 3 and a_1 = 5, what coefficients α_1 and α_2 satisfy this in the previous multiple root example?

   Hint

   Substituting initial conditions helps find unknown constants.