All right this question has been bothering me, I sort of stumbled upon it myself and have tried it once before but with no success, it is the reason I signed up for this forum. I apologise if this question has been posted in the wrong section of this forum, it doesnt really have a place to go.
Example using numbers:
If you want the square root of 20 then you 'guess' the value as being 4 and the process to solve the square root is as follows.
(4+20/4)/2 = (36/4)/2 = 36/8 = 9/2 --> This is our first approximation.
Next we use the first approximation as our guess and continue with the same process:
(9/2 + 20 / (9/2) ) / 2 = (161/18)/2 = 161/36 ---> this is our second approximation.
Continuing with this pattern for infinite terms will yield exactly the sqare root of 20, our second approximation is already close:
161/36 = 4.472222222...
root20 = 4.4721359549996
If we generalise this process and call our starting 'guess' number as A and the number we are trying to square root as X then the n+1 term can be written in terms of the nth term.
T(n+1) = (T(n) + X/T(n)) / 2
My question is, and the question ive been trying to solve for a while now is how can I express the nth term relative to the first term? Is it even possible? I want something along the lines of:
T(n) = some formula involving T(1)