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?