16.3.1. Recurrence Condition

Try these first

1. 1.

   Define what a recurrence relation is.

   Hint

   Think about how each term in a sequence might relate to earlier terms.

2. 2.

   What is S(1) if we define S(n) as the count of strictly increasing sequences?

   Hint

   Consider the sequence with only one element.

3. 3.

   What defines a strictly increasing sequence?

   • Each term is less than the previous
   • Each term is greater than the previous
   • Each term can be equal

   Hint

   Consider the meaning of 'strictly'.

4. 4.

   True or False: The initial conditions for S(n) are always the same for all sequences.

   • True
   • False

   Hint

   Think of sequences that start differently.

5. 5.

   If you have S(5), how would you compute it based on previous terms? Show your work.

   Hint

   Ensure you keep track of how categories previously derived affect your counts.

6. 6.

   Explain how modifying the sequence rules can affect S(n). Provide a hypothetical situation.

   Hint

   Think about what these changes might do to valid selections.