15.7. Theoretical General Form of the Solution

Try these first

1. 1.

   Define characteristic roots.

   Hint

   Think about the roots of a polynomial.

2. 2.

   What is a homogeneous recurrence equation?

   Hint

   Consider the structure of terms in the recurrence.

3. 3.

   What does the general form of a solution involve when roots are distinct?

   • Polynomials of degree k
   • Linear combination of roots raised to n
   • Only one constant term

   Hint

   Think of the structure provided from roots.

4. 4.

   True or False: Repeated roots do not influence the solution's structure.

   • True
   • False

   Hint

   Reflect on the earlier discussion about root multiplicity.

5. 5.

   For the recurrence relation T_n = T_{n-1} + T_{n-2}, identify the roots and derive the general solution.

   Hint

   Start with the characteristic polynomial and factor it.

6. 6.

   Given T_n = 2 T_{n-1} + T_{n-2} with the initial conditions T_0 = 1 and T_1 = 2, find T_n for n = 5.

   Hint

   Work through the constants based on initial value equations.