Ho do you prove this using Pascal sequences?

Draw your Pascal triangle.

Line n of the triangle contains, from left to right, the coefficients (n r).

The "diagonals" from top right to bottom left correspond to values of r constant.

You know that an element of line n+1 of the triangle is equal to the sum of the 2 elements in the line above.

So (n+1 r) = (n r-1) + (n r)

and (n+1 r+1) = (n r) + (n+1 r)

Now apply the same logic to line n+2

(n+2 r+1) = (n+1 r+1) + (n+1 r)

which gives you your result.