proving that a number is a perfect square I need help with this problem:
$K$ is equal to $111\ldots111 - 22\ldots22$ where $1$ is used $2n$ times and $2$ is used $n$ times. Prove that $K$ is a perfect square.
It is true, but I can't find a way to prove it. The only interesting thing I've found is that when $n$ is $1, K=3^2$, when $n$ is $2, K=33^2$ and so on.
I tried to prove it using induction but wasn't able to find a way to do it, so your help would be really appreciated.

The number that you are looking at is $$N = (10^{2n-1}+10^{2n-2}+\ldots+10+1)-2(10^{n-1}+10^{n-2}+\ldots+10+1)$$
Summing the GPs,
$$N = \frac{10^{2n}-1}{9}-2\frac{10^n-1}{9} = \frac{10^{2n}-2\times10^n +1}{9} = \bigg(\frac{10^n-1}{3}\bigg)^2$$
$\frac{10^n-1}{3}$ will always be of the form $333\ldots$ So, $N$ is a perfect square for all $n$.
