14.8. Proof of Theorem - Part 2

Try these first

1. 1.

   What is the general form of a linear homogeneous recurrence relation?

   Hint

   Look for the structure where the n-th term relates to its predecessors.

2. 2.

   Define characteristic roots in the context of recurrence relations.

   Hint

   Think about how these roots connect to the structure of the recurrence.

3. 3.

   What does the characteristic equation help us identify?

   • The first term of the sequence
   • The characteristic roots
   • The final term of the sequence

   Hint

   Recall its purpose in the solution process.

4. 4.

   True or False: All roots in a characteristic equation must be distinct.

   • True
   • False

   Hint

   Consider instances where roots overlap.

5. 5.

   Consider the recurrence relation S(n) = 2 * S(n-1) + 3 * S(n-2) with S(0) = 1 and S(1) = 4. Find the characteristic roots and derive the general form of the solution.

   Hint

   Start with the characteristic equation before substituting in initial conditions.

6. 6.

   Given a recursion with repeated roots, S(n) = S(n-1) + S(n-2) with initial conditions S(0) = 1 and S(1) = 1, show how you would find the nth term.

   Hint

   Make sure to label your roots correctly when handling repeated instances.