14.2. Definition of Linear Homogeneous Recurrence Equations

Try these first

1. 1.

   Write the general form of a linear homogeneous recurrence equation of degree 2.

   Hint

   Consider how n-th term relates to its previous terms.

2. 2.

   What is the Fibonacci recurrence relation?

   Hint

   Look at how each term is generated from the two previous terms.

3. 3.

   What is a linear homogeneous recurrence relation?

   Hint

   Recall the general form presented in class.

4. 4.

   True or False: The characteristic equation is always a quadratic for degree 2 recurrence relations.

   • True
   • False

   Hint

   Think about the degree of the recurrence.

5. 5.

   Given the recurrence relation a_n = 4a_{n-1} - 4a_{n-2} with initial conditions a_0 = 2 and a_1 = 10, find the explicit formula and calculate a_2.

   Hint

   Substitute n = 0 and n = 1 into the general form to create equations.

6. 6.

   Analyze the recurrence relation a_n = a_{n-1} + a_{n-2} and prove that it behaves like the Fibonacci sequence.

   Hint

   Look at how each term forms by summing the two preceding terms.