13.8. Uniqueness of Solutions for Recurrence Equations

Try these first

1. 1.

   What is a recurrence equation?

   Hint

   Think about sequences and how their terms relate.

2. 2.

   Define initial conditions in the context of recurrence relations.

   Hint

   Consider what you need to start calculating a sequence.

3. 3.

   What defines a linear homogeneous recurrence equation?

   • It has multiple solutions.
   • It does not include constant terms.
   • It is always quadratic.

   Hint

   Focus on the structure of the equation.

4. 4.

   True or False: All recurrence equations have one and only one solution.

   • True
   • False

   Hint

   Think about what happens without fixed starting points.

5. 5.

   Given the recurrence relation T(n) = 4T(n-1) - T(n-2) with initial conditions T(0) = 1, T(1) = 2, derive T(5).

   Hint

   Work step by step, substituting previous results until you find T(5).

6. 6.

   Prove that for the recurrence T(n) = 3T(n-1) with T(0) = 1, the sequence converges to a unique solution by induction.

   Hint

   Use the assumption to show how it holds for the subsequent term as well.