Pat’s Blog had a post about infinite radicals, which looked at . The problem is just as easy to solve for if you replace the 3 with .
Note that this technique is almost identical to how one solves an infinite geometric series, so would make a good challenge problem right after introducing geometric series. This could be appropriate for a math team recreational puzzle.
One can obviously generalize to roots:
And of course one would have to discuss when the formal manipulation makes sense. For example, leads to the “solution” , which is clearly wrong. It might be worth some calculator time for the kids to play with different values of to see when the infinite radical makes sense. The value of for should be a familiar number.
One could also solve the quadratic to see what values takes on for different values of , and look for a relationship between the parts of the quadratic formula and the solutions to the infinite radical chain.