Originally posted by Swlabr
You use previous results, while I just hit it with a stick.
Seeing a hint for an elegant solution would be nice, Mr 113.
The ideal solution uses no previous results and hits it with a pretty stick 🙂
I guess what OP wants is:
a(n+1)^2 = 2 + a(n)^2 + 1/a(n)^2 > 2 + a(n)^2
So that by the same telescoping as before, a(n+1)^2 > 2*99 + 1.
Just squaring it is certainly short, but it somehow obscures the issue. In general, nonlinear difference equations like this one are hard to solve -- they're about as hard to solve as the corresponding nonlinear differential
equations. In this case, though, the equation a(n+1) - a(n) = 1/a(n) corresponds to the equation dy/dx = 1/y, which just happens to be
easy to solve, and the relationship between a(n) and 2n+1 reflects this. Hiding in the malarkey I posted yesterday is also the upper bound a(100)^2 < 2*100 + 1.