This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
This can be solved by noting a more general formulation:
x= \sqrt{ax+(n+a)^2 +x\sqrt{a(x+n)+(n+a)^2+(x+n) \sqrt{\mathrm{\cdots}}}} \, Setting this to F(x) and squaring both sides gives us:
F(x)^2 = ax+(n+a)^2 +x\sqrt{a(x+n)+(n+a)^2+(x+n) \sqrt{\mathrm{\cdots}}} \, Which can be simplified to:
F(x)^2 = ax+(n+a)^2 +xF(x+n) \, It can then be shown that:
F(x) = {x + n + a} \, So, setting a =0, n = 1, and x = 2:
3 = \sqrt{1+2\sqrt{1+3 \sqrt{1+\cdots}}}. \,