Suppose x_n = (a+x_(n-1))^0.5, where x_(n) means x sub nand let x_1 = a^0.5?

Question

Is there any easy way to evaluate x_k exactly?

Answer 1

I believe you can work this out this way. First find x_2

x_2 = (a+x_1)^.5; x_1 = a^.5 so x_2 = (a + a*.5)^.5 = a^.5*(1+a^.25)^.5

Then find x_3 = (a+x_2)^.5; put in x_2 from above.

Soon a pattern wil emerge which you could write in general terms for a_k.

I may work on this some more and post a more complete answer, but I think you should be able to do it.

EDIT: I have made an attempt at a general formula based on a repetitive pattern. You can find it here:
http://img309.imageshack.us/img309/6682/recursivepatternen4.png
I'm not sure if there is a mathematical notation to express that.

Answer 2

I can see the value of x and n ,but where it the value of k.I think that something is missing in here.