14.6. Theorem Statement

Try these first

1. 1.

   Define a linear homogeneous recurrence equation.

   Hint

   Look at how the n-th term is constructed from prior terms.

2. 2.

   What is a characteristic equation?

   Hint

   Recall how you form it from the recurrence relation.

3. 3.

   What form does the n-th term of a recurrence equation take when the roots are distinct?

   • a_n = r^n
   • a_n = αr_1^n + βr_2^n
   • a_n = c_1 + c_2n

   Hint

   Think about how we construct the sequence based on distinct roots.

4. 4.

   True or False: An initial condition can determine multiple sequences that fit a single recurrence relationship.

   • True
   • False

   Hint

   Consider the implications of having the same starting point.

5. 5.

   Given a recurrence relation with the characteristic equation $r^2 - 5r + 6 = 0$, find the general form of the n-th term and particular values for $a_0 = 2$, $a_1 = 3$.

   Hint

   Start by finding the roots, then use the initial terms in your general form.

6. 6.

   Describe how the distinct roots could influence the behavior of a recurrence relation in terms of convergence.

   Hint

   Comparing growth rates can help determine which terms will dominate as n increases.