16.1.5. Initial Conditions

Try these first

1. 1.

   Define a valid sequence in your own words.

   Hint

   Think about the starting and ending numbers and the rules of increasing order.

2. 2.

   What is a recurrence relation?

   Hint

   It connects current values with past values.

3. 3.

   What does a valid sequence consist of?

   • Starts and ends with numbers
   • contains any numbers in between
   • Starts with one number and ends with another in strictly increasing order
   • A random sequence of numbers

   Hint

   Consider the definition of valid sequences.

4. 4.

   Is the following statement True or False? 'Initial conditions are merely optional in recurrence relations.'

   • True
   • False

   Hint

   Think about why we need starting points for sequences.

5. 5.

   Suppose you have a sequence defined recursively. If s(1) = 1, s(2) = 1, and s(n) = s(n-1) + s(n-2) for n > 2, what is s(5)?

   Hint

   Work backwards from the values you know.

6. 6.

   Create a sequence ending with '5' and starting with '1', listing all the valid strictly increasing sequences.

   Hint

   Count valid combinations based on the defined rules.