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.