(* Newton's Method *)
f[x_, a_] := x^2 - a;
nextterm[old_, a_] := (old^2 + a) / (2 * old);

nextterm[2,2]

previous = 2;
num=3;
Print["We are finding ", Sqrt[num]," = ", SetAccuracy[Sqrt[num],40]];
Print["Initial Guess: Term 0: ", previous, " = ", SetAccuracy[previous, 15]];
For[j = 1, j ≤ 5, j++,
{
Print[" "];
previous = nextterm[previous, num];
Print["Term ", j, ": ", previous, " = ", SetAccuracy[previous,15]];
Print["Difference to ", Sqrt[num], " = ", SetAccuracy[Abs[previous-Sqrt[num]],40]];
}];