The difference between two successive square numbers is the sum of their respective square roots:
(x + 1)^2 − x^2 = x2 + 2x + 1 − x2 = x + (x + 1)
So if you know that 10*10=100 then you can get 11^2 as:
100 + 10 + 11 = 121
Now, you can get 12^2 as:
121 + 11 + 12 = 144
And for 13^2:
144 + 12 + 13 = 169
For 14^2:
169 + 13 + 14 = 196.
Now, lets try it for 999999^2:
We know that 1000000^2 is 1000000000000. So:
999999^2 + 999999 + 1000000 = 1000000000000
999999^2 = 1000000000000 - 1000000 - 999999 = 1000000000000 - 2000000 + 1 = 999998000001