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.
Virtually the very first “math” problem I got interested in involved a 7th grade homework problem in 2005 that a colleague at work said her son had been given. I ended up commenting and helping on a number of further problems, which gave me some insight into the state of current public school teaching in mathematics. It was both encouraging and discouraging at the same time. I will join the math education commentary at a later date.
The problem was not that bad: What is the largest power of 2 that divides 800! without a remainder? (where “!” means “factorial”, for example, 5! = 5 x 4 x 3 x 2 x 1). I solved it in my usual pedestrian way. I showed it to a friend of mine (an algebraist!) and he of course had a nifty approach. He showed it to a colleague of his at NSF (a physicist) and he had the niftiest solution of all! (Most humbling.) See the Power of 2 Problem.