Question

find the value of x:
\[x = \sqrt{2 + \sqrt{2 + \sqrt{2 +...}}}\]

Answer 1

\[x=\sqrt{2+x}\]\[x^2=2+x\]\[x^2-x-2=0\]\[(x-2)(x+1)=0\quad\Rightarrow\quad x=2\]

Answer 2

wait really?

Answer 3

nicely done!

Answer 4

wow yea... sickk!!

Answer 5

nikvist is right, but for any rigorous mathematician, you have to establish the convergence of \( x = \sqrt{2 + \sqrt{2 + \sqrt{2 +...}}} \)

Related discussion in M.SE : http://math.stackexchange.com/questions/61012/nested-square-roots

http://math.stackexchange.com/questions/61048/definition-of-convergence-of-a-nested-radical-sqrta-1-sqrta-2-sqrta-3