15.1. Recap of Last Lecture

Try these first

1. 1.

   Define a linear homogeneous recurrence equation.

   Hint

   Think about sequences like Fibonacci.

2. 2.

   What is the purpose of the characteristic equation?

   Hint

   It helps us find the roots related to the recurrence relation.

3. 3.

   Which of the following is true about the characteristic equation?

   • It is unrelated to the recurrence relation.
   • It helps identify the roots needed to determine solutions.
   • It is always quadratic.

   Hint

   Think about its role in helping form sequences.

4. 4.

   True or False: All roots of a characteristic polynomial must be distinct.

   • True
   • False

   Hint

   Consider cases with Fibonacci-like sequences.

5. 5.

   Consider the recurrence relation T(n) = T(n-1) + 2T(n-2). Find the characteristic roots and solve the equation. Discuss what happens when modifying coefficients.

   Hint

   Factor the characteristic polynomial.

6. 6.

   Create your own recurrence relation with at least one repeated root. Solve for the nth term and explain your steps.

   Hint

   Focus on the polynomial degree based on root multiplicities.