e.g. number $5$.
$5^2=25;$ If I add up the first $4$ even numbers and a half of the $5$th even number, we get $25$. $2+4+6+8+5=25$
So generally speaking, the sum of first $n-1$ even numbers and the half of $n$th number is equal to $n^2$.
I am interested if this is used somewhere? If not, could it be used? Thanks!
The sum of the first $n-1$ positive integers is $n(n-1)/2$ by a well known formula. Your sum is this multiplied by $2$, plus $n$, which is $n^2-n+n=n^2$. This is a relatively trivial observation.