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.

q.e.d