1.7.1—Proof by induction

To prove P(n) for all integers n≥n₀, prove P(n₀), assume P(k) for an arbitrary k≥n₀, then use that assumption to prove P(k+1).

The assumption is used only for the arbitrary case, not asserted for every value without proof. State the domain and simplify the target P(k+1) until the expression contains the inductive hypothesis.

For 1+2+…+n=n(n+1)/2, the base case n=1 is true; adding k+1 to the assumed sum k(k+1)/2 gives (k+1)(k+2)/2.

Checking many numerical cases is not induction, and proving P(k)→P(k+1) without a valid base case proves nothing about the starting chain.