X_n = 2·X_n-1·Y_n-1/(X_n-1+Y_n-1)

Y_n = sqrt(X_n·Y_n-1)

prove:

a) that X_n decreases to X and Y_n increases to y as n --> infinity for some x and y in R.

b) that x=y and that value is pi

thanks

https://www.cut-the-knot.com/ctk/Magic.shtml

This will give you a clue as to why x is decreasing, y is increasing and why in the limit x = y. Off the top of my head I can't tell you right away why both will be equal to Pi.

Regards,
Alexander Bogomolny