Without the aid of a calculator or any other computing device,
determine the value of the infinite expression 
.
Solution by Robert Freund, Alma High School
Let 
, and square
both sides; Then
So, x = -2 or x = 3, but the solution to a radical can’t
be negative. Therefore, 
.
NOTE: If k is any positive real number, then we can
evaluate 
as follows:
Let 
; then we
obtain the equation, 
, whose solution (using the quadratic formula) is 
.
So, in general, 
.
Also solved by: Pranav Moudgil (Detroit Country Day H.S.); Alex Chiao (Ann Arbor Huron H.S.); Eva Dou, Alex Bordyukov (Grosse Pointe South H.S.); Andrew Jeanguenat (De La Salle Collegiate H.S.); Visakh Ponnuru, Kelcie Daniels (Forest Hills Northern H.S.); Jacob Arnett (Alma H.S.); Ikbol Ahmedov, Jake Nugent (Benzie Central H.S.); Eric Anderson, Becky Pimm, Drew Reyelts, Lindsey Westerhof, Marc Walker, Max Dixon, Mischa Dylewski, Andrew Rennaker, Jay Yang (Forest Hills Central H.S.); Shaun Pezeshki (West Bloomfield H.S.); Connie Shi (Okemos H.S.); Hao Li (East Lansing H.S.); Robert Harrison (Lansing Catholic H.S.); Erica Wozniak (Marquette Sr. H.S.); John Abraham, Luke Fischer, Kayla Kalmbach (East Grand Rapids H.S.); Whitney Askew, Walter Wang, Alex Berk (Plymouth H.S.);Katelyn Evans (Forest Hills Northern H.S.); Kiran Sebastian, Alex Timm (Saginaw Heritage H.S.); Armila Francis, Patrick Lutz (Plymouth Christian Academy); Lindsay Poremba, Sarah Peterson (H.S. not provided); Husam Alghanem (Grand Blanc H.S.); and Asra Shaik (Novi H.S.).