Actually this one formula has many beautiful and different ways of being deduced. It probably also has been asked a few times before on this website? The way most people first see, along with the story of the young Gauss figuring it out when he was 11 years-old or something.
It goes like this : First, write the sum of the first $n$ integers, once forwards and once backwards.
it is not too hard to see that if you sum together one number from the top row with one number from the bottom row you will always get the result $n+1$, but then you will be summing everything twice, which means that the total sum is $n(n+1)/2$.
As far as I know there isn't really a "general" way of finding formulas for arbitrary sums, the strategies to find them are usually different in each case.