Here is surprising problem from the 1875 The Analyst
“81. By G. W. Hill, Nyack Turnpike, N. Y. — Prove that, identically,
By “identically” the proposer means for all n = 1, 2, 3, ….
See the Surprising Identity
(Update 8/20/2021) James Propp at his website has an informative, extensive article on mathematical induction and its variations.