13.5. Solving Recurrence Equations

Try these first

1. 1.

   Define a recurrence equation.

   Hint

   Look for terms that express future values based on past values.

2. 2.

   What is S(1) for bit strings avoiding '00'?

   Hint

   Consider all possible single digit strings.

3. 3.

   What defines a recurrence equation?

   • An explicit formula
   • A method to solve algebraic equations
   • An equation relating terms of a sequence

   Hint

   Think of it as a definition involving sequences.

4. 4.

   True or False: Recurrence relations can only have a single solution.

   • True
   • False

   Hint

   Consider scenarios of supplying initial terms.

5. 5.

   Prove that for any recurrence relation of the form T(n) = aT(n-1) + b, where a ≠ 0, you can derive a formula.

   Hint

   Start with base cases and build upwards.

6. 6.

   Demonstrate the uniqueness of solutions in a linear homogeneous recurrence relation when given initial conditions matching its degree.

   Hint

   Reflect on how initial terms dictate the future terms.