19.2.1. General Form of Linear Non-Homogeneous Recurrence Equations

Try these first

1. 1.

   What is the general form of a linear non-homogeneous recurrence equation?

   Hint

   Listen for components regarding previous terms and an external function.

2. 2.

   What does F(n) represent?

   Hint

   Think about the influence exerted outside the sequence.

3. 3.

   Which term describes the part of the recurrence equation that contributes external factors?

   • Homogeneous Part
   • Non-Homogeneous Part
   • Characteristic Equation

   Hint

   Look for the component that does not derive from previous terms.

4. 4.

   True or False: The associated homogeneous relation is obtained by maintaining F(n).

   • True
   • False

   Hint

   Remember that we're simplifying the equation.

5. 5.

   Find the general form of a linear non-homogeneous recurrence equation with F(n) = 5n^2. What if this polynomial does not share roots with the characteristic polynomial?

   Hint

   Consider how the degree of F(n) influences your particular solution guess.

6. 6.

   Given F(n) = 3^n, identify how a solution changes if this component becomes a characteristic root with multiple occurrences (say, a root of 3 with multiplicity 2)?

   Hint

   Reflect on adjustments needed whenever roots are matched.