16.7. Combinatorial Proof of Identity

Try these first

1. 1.

   Define a valid sequence in your own words.

   Hint

   Think about the properties that define a sequence.

2. 2.

   What does the function $S(n)$ represent?

   Hint

   Consider the meaning of sequences in increasing order.

3. 3.

   What is the form of the recurrence relation derived in this section?

   • S(n) = S(n-1) + S(n-2)
   • S(n) = 2S(n-1)
   • S(n) = S(n-1) + S(n)

   Hint

   Recall the formula discussed during the session regarding the last term.

4. 4.

   True or False: The number of valid sequences with $n=1$ is greater than with $n=2$.

   • True
   • False

   Hint

   Think back to the initial conditions we confirmed.

5. 5.

   Given $n=4$, calculate all differing valid sequences and confirm $S(4)$.

   Hint

   Consider each element's position and how many can fit in the sequence.

6. 6.

   Explore how introducing another initial condition alters the recurrence.

   Hint

   This involves checking sequences' sizes and corresponding counts.