Consider the famous Pascal triangle, purposefully drawn in a somewhat lopsided way:

1
1  1
1  2  1
1  3  3  1
1  4  6  4  1
1  5 10 10  5  1
1  6 15 20 15  6  1
......................

Start at any number, and draw a line at 45 degrees, from bottom left to top right. (For example, if you chose the first "4" of the fifth row, the diagonal would also include a "1" and a "3")

How much do the numbers in such a line sum? Why? Can you prove it?

(In reply to solution by Charlie)

I agree, but you must add the fact that the two first diagonals add to 1, and then you do have the Fibonacci sequence. I found the proof to be clearer, after adding zeroes all around the triangle.

Posted by e.g. on 2004-03-15 14:22:39