X1 = 2, X2 = 1.75, X2 = 1.732142857, X4 = 1.73205081 X5 = 1.732050808, X6 = 1.732050808 , X7 1.732050808 X8 = 1.732050808 and so on

We notice that when X_n = 1.732050808, so is X_n+1. Squaring these terms we get X₁² = 4, X₂² = 3.0625, ... , X₅² = 3 and the rest of the other terms are the same!!

This implies that when X_n » Ö3 so is X_n+1 and the values of X_n tend to the limit Ö3. This special property can easily be proven. Assume that the limit exists, so X_n+1 = X_n = X, then solve the equation

X = 1 2 æ ç è X + 3 N ö ÷ ø

If we test it for N = 3, we see that X₂₉ = 1.44224957, which is what the calculator gives for the cube root of 3. Testing it for N = 8, we get X₁ = 2, which is the right answer.