14.5. Characterization of Sequences

Try these first

1. 1.

   What is a linear homogeneous recurrence equation?

   Hint

   Look for terms that establish a relationship between sequence terms.

2. 2.

   Define characteristic roots.

   Hint

   Think of them as the values essential for the general solution.

3. 3.

   What is the characteristic equation for the recurrence $a_n = a_{n-1} + a_{n-2}$?

   • $r^2 - r - 1 = 0$
   • $r^2 - 2 = 0$
   • $r^2 + 1 = 0$

   Hint

   Think of how each previous term influences the next.

4. 4.

   True or False: A linear homogeneous recurrence can have non-constant terms.

   • True
   • False

   Hint

   Recall what it means for an equation to be homogeneous.

5. 5.

   Given the recurrence $a_n = 2a_{n-1} + a_{n-2}$ with initial conditions $a_0 = 1$, $a_1 = 2$, derive and solve for $a_2$ and $a_3$. Show each step.

   Hint

   Remember to substitute the previous terms into the equation.

6. 6.

   Prove that if an equation has distinct roots, the form $a_n = \alpha r_1^n + \beta r_2^n$ holds true. Use an example.

   Hint

   You might require the quadratic formula to find your roots.