The "statement" in a mathematical induction proof is called the inductive hypothesis. In the last example the inductive hypothesis was

P(n) = 1 + 2 + 3 + ... + n = n(n + 1)/2

the rule of inference describing mathematical induction is

[P(1) n (P(n)P(n+1))] n P(n)

(Actually this is only correct when n is a positive integer – if n comes from another set then this would need to be modified.)