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.