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Example

We prove by induction on n

forall n in N: (sum_{1 <= i <= n} i) = (n+1)n/2

The induction base holds because

(sum_{1 <= i <= 0} i) = 0 = (0+1)*0/2.

We take arbitrary n in N and assume

(1) (sum_{1 <= i <= n} i) = (n+1)n/2.

We have to show

(2) (sum_{1 <= i <= n+1} i) = ((n+1)+1)(n+1)/2.