When the positive integers are arranged in order, filling in the successive diagonals of an infinite grid from top to bottom, as shown, the integer 41 is in the (5,5) spot. What integer would we see in the (10,20) spot if the rest of the grid were visible?
+128408 Using the sum of differences
1 2 4 7 11
1 2 3 4
1 1 1
We have two rows of non-zero results
The polynomial generating this series is ax^2 + bx + c
So we have this system
a + b + c = 1
4a + 2b + c = 2
9a + 3b + c = 4
Solving this system gives a = 1/2 b = -1/2 c = 1
So....the 10th term in this row = 10^2 / 2 - 10/2 + 1 = 50 - 5 + 1 = 46
Also in the 10th column the terms are
46 57 69 82 96 111
11 12 13 14 15
1 1 1 1
And we have this system
a + b + c = 46
4a + 2b + c = 57
9a + 3b + c = 69
Solving this we get a = 1/2 b = 19/2 c = 36
Then the 20th term is 20^2/2 + (19)(20)/2 + 36 = 426
So position (10 , 20) = 426
