1. ## pascal sequences

This problem uses Pascal sequences but I don’t know how to insert mathematical symbols, so I’ll use (n r)

A sequence F(nth term) is constructed from terms of Pascal sequences as follows:
F(0 term) = (0 0)
F(1st term) = (1 0) + (0 1)
F(2nd term) = (2 0) + (1 1) + (0 2)
and in general
F(nth term) = (n 0) + ((n-1) 1) + … + (1 (n-1)) + (0 n)

Use the Pascal property (n r) + (n (r+1)) = ((n+1) (r+1)) to explain why F(3rd term) + F(4th term) = F(5th term) and F(4th term) + F(5th term) = F(6th term)

OK so I was wondering what would be an appropriate way to do this, ‘proving’ these things would be pretty easy but ‘explaining’ is something entirely different. From a teacher’s point of view, what would be a satisfactory way for a student to answer this? Would it be enough to show (in the first case) that (4 4)=(5 0) + (4 1) + (3 2) + (2 3) + (1 4) + (0 5) by direct calculation?

2. OK so I was wondering what would be an appropriate way to do this, ‘proving’ these things would be pretty easy but ‘explaining’ is something entirely different.
I would say that proving is the ultimate form of explanation (to a point when it sometimes becomes an obfuscation).

Would it be enough to show (in the first case) that (4 4)=(5 0) + (4 1) + (3 2) + (2 3) + (1 4) + (0 5) by direct calculation?
In my opinion, sure. In modern interactive theorem provers (programs that help people prove mathematical theorems by doing all bookkeeping and performing small reasoning steps automatically), this is called "proof by computation", or Poincaré principle. It means that if E1 and E2 are two arithmetic expressions that evaluate to the same number, say, 2, then one does not even have to prove that E1 = E2. To the computer, E1 = E2 is the same as 2 = 2 because it evaluates expressions behind the scenes.

I agree, of course, that a proof by computation in this case would probably not advance your understanding of why the sums of the diagonals in the Pascal's triangle are the Fibonacci numbers. However, the role of a proof is to be strict and utterly convincing. Though good proofs are always illuminating, not all proofs are because human brain "understands" things through a different mechanism than formal proofs.

From a teacher’s point of view, what would be a satisfactory way for a student to answer this?
Ideally, one should strive more to get to the bottom of things than to please a teacher. Of course, it's easy for me to say this when I am no longer in school I would say that the answer to your question depends on how much you want to understand what's going on and how much effort you are willing to spend. A proof by computation is sufficient and you can be sure that it is indeed a proof. If you want a general proof, look at this explanation and in Wikipedia.