15.8. Example Application of General Formula

Try these first

1. 1.

   What is the characteristic equation for a_n = 4a_{n-1} + 4a_{n-2}?

   Hint

   Derive the equation based on the recurrence relation.

2. 2.

   Define a homogeneous recurrence relation.

   Hint

   Think about the form of the equation.

3. 3.

   What type of equation derives from a linear homogeneous recurrence relation?

   • Characteristic Equation
   • Quadratic Equation
   • Polynomial Equation

   Hint

   Remember the definition of a characteristic equation.

4. 4.

   True or false: Repeated roots require polynomial terms multiplied by n.

   • True
   • False

   Hint

   Think about how repeated roots modify the general solution.

5. 5.

   Given a recurrence relation a_n = -2a_{n-1} - 5a_{n-2}; find and analyze the characteristic roots.

   Hint

   Use the quadratic formula to find the roots.

6. 6.

   Consider the recurrence relation a_n = a_{n-1} + a_{n-2} with initial conditions a_0 = 2, a_1 = 3. Find the specific sequence.

   Hint

   You can set up equations from known initial values to solve for the sequence coefficients.