1.7.2—Inductive conjecture

A conjecture is a statement suggested by patterns in initial cases. Induction can then prove it if the base case and inductive implication are valid for the stated domain.

Use small values to identify the likely formula, factorisation or invariant, but do not treat pattern matching as evidence of universal truth. Check parity, starting index and exceptional values before writing the proof.

The first differences of a sequence may suggest a quadratic formula. Test several terms, formulate P(n), then prove P(n₀) and P(k)⇒P(k+1) rather than extrapolating from the table.

A pattern that works for ten values can fail at the next; conjecture and proof are different learning jobs.