154. Theorem Statement for Distinct Roots

Try these first

1. 1.

   What is a characteristic equation?

   Hint

   Think of how we find solutions to polynomials.

2. 2.

   Define distinct roots in the context of recurrence relations.

   Hint

   What would happen if roots were the same?

3. 3.

   What is the general form of a recurrence relation with two distinct roots?

   • a_n = α₁r₁ⁿ + α₂r₂ⁿ
   • a_n = α(r₁ + r₂)ⁿ
   • a_n = (r₁ + r₂)ⁿ

   Hint

   Remember the format with constants.

4. 4.

   True or False: Distinct roots result in a unique sequence for any initial values.

   • True
   • False

   Hint

   Recall how distinct roots behave.

5. 5.

   Given the recurrence relation a_n = 5a_{n-1} - 6a_{n-2}, find the characteristic roots and express the general solution.

   Hint

   Think about how to factor the quadratic equation.

6. 6.

   Derive the general term for a recurrence relation a_n = a_{n-1} + a_{n-2} with given initial conditions a_0 = 1, a_1 = 1.

   Hint

   Remember to equate using the initial conditions.