Question

Please simplify:

Answer 1

\[\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+.....}}}}\]

Answer 2

\[(1+\sqrt{5})/2\] = 1.618

Answer 3

thanks! but how'd you come up with that?

Answer 4

let\[x = \sqrt{1+\sqrt{1+\sqrt{1+...}}}\]
Therefore you can write it in the form
\[x=\sqrt{1+x}\]
Take a moment to think about that, substitute the x= part into the x on the right side of the equation and repeat indefinitely.
Therefore we can write this as
\[x^2 = 1+x\]
rearange to get
\[x^2 - x - 1 = 0\]
solve the quadratic using your favorite method and BAM done! :D
\[x = (1+\sqrt{5})/2\] or \[x=(1-\sqrt{5})/2\]
In this case we only consider the positive root as it is going to be under the square root and we cant get real values for the square root of negative numbers.

Hope that's helpful :)

Answer 5

ill try to understand this. thanks you very much!! :D

Answer 6

the hard part is not finding this number. the hard part is proving that it converges so some limit

Answer 7

this number is called "golden mean" or "golden ratio" or simply "phi" and you can read about it many places, for example here
http://en.wikipedia.org/wiki/Golden_ratio