I understand the proof by induction, however I would like to understand the intuition behind why when doing the direct proof we simply layout the the sum forwards and backwards and add the two as such:
$$S= 1 + 2 + 3 + ... + (n-1) + n $$ $$S = n + (n - 1) + (n - 2) + ... + 2 + 1$$ $$2S = (n + 1) + (n + 1) + (n + 1) + ... + (n + 1) + (n + 1) = n(n+1)$$ (then just divide by 2)
How do we figure out that doing such procedure will give us the appropriate formula? What's the general essence of such reasoning that can help us to reason other proofs of different nature?