14.4. Demonstration with Degree 2 Recurrence Equations

Try these first

1. 1.

   What is a general characteristic equation for degree 2?

   Hint

   Think about how to express the linear combination.

2. 2.

   Can the characteristic roots be the same for degree 2 equations? If yes, what does it imply?

   Hint

   Recall what happens when roots repeat in polynomials.

3. 3.

   What type of equation defines linear homogeneous recurrence equations?

   • Quadratic Equation
   • Linear Combination
   • Polynomial of Degree 2

   Hint

   Think about how many previous terms it considers.

4. 4.

   True or False: The Fibonacci sequence is a linear homogeneous recurrence relation.

   • True
   • False

   Hint

   Consider how the sequence is defined!

5. 5.

   Given the recurrence relation a(n) = 2a(n-1) + 3a(n-2) with a(0) = 1 and a(1) = 2, find the explicit formula for a(n).

   Hint

   Set up the characteristic equation from the recurrence relation.

6. 6.

   If a sequence is defined as a(n) = a(n-1) + 5a(n-2) and you know a(2) = 7, a(3) = 12, derive the next term a(4).

   Hint

   Think about how each term builds on the last two.