16.1.2. Compact Recurrence Condition

Try these first

1. 1.

   What is a strictly increasing sequence?

   Hint

   Think of the order of numbers.

2. 2.

   Write down the first two terms for S(n) when n = 1 and n = 2.

   Hint

   Think about what sequences you can form.

3. 3.

   What defines a strictly increasing sequence?

   • All terms are equal
   • Each term is less than the previous
   • Each term is greater than the previous

   Hint

   Visualize the order of numbers.

4. 4.

   True or False: Initial conditions are unnecessary for solving recurrence relations.

   • True
   • False

   Hint

   Think about where you’d start in a sequence.

5. 5.

   Create a sequence starting with 1 and following stringent increasing order rules for n = 5, representing the sequences valid within those terms. How many such sequences can you generate?

   Hint

   Use your relationship derived from earlier S values.

6. 6.

   Propose a different compact recurrence relationship if you allowed sequences to include numbers in non-strict order (where values can repeat).

   Hint

   Think of combinations of terms rather than strict orders.