A simple proof I can think of is
(a+b+c)²=a² + b² + c² + 2ab + 2bc + 2ac
When n=1 in your case,
a=10000, b=200, c=1
The resultant is 100000000 + 40000 + 1 + 4000000 + 400 + 20000 = 104060401.
This will continue as n increases.
For example, for n=2,
a=1000000, b=2000, c=1.
The resultant would be 1000000000000 + 4000000 + 1 + 4000000000 + 4000 + 2000000 = 1004006004001.
It can be seen that the other digits are not affected because of the number of zeros in a and b.
Hence, for any value of n, the resultant is a perfect square.