The Babylonian method for square roots is based on the simple fact that sqrt(x)*sqrt(x)=sqrt(x^2)=x.
So sqrt(x)=x/sqrt(x). If we approximate to sqrt(x) and call it x1, then we can write x2=x/x1 approximately. The actual square root of x will lie somewhere in between x1 and x2. The midpoint of these two values is the average (x1+x2)/2. Call this value x3. This will be a more accurate approximation than x1. So we start with x3 and x4=x/x3. The average of x3 and x4 is (x3+x4)/2=x5. And so on.
Now the question. x=115. We could start with x1=10, so x2=11.5 and x3=10.75 (approx 10.8). Then x4=115/x3=10.7 (approx) and x5=10.72. So the progressive approximations to sqrt(115) are: 10,10.75,10.72380529. The question isn't clear as to where the blanks are, but it would appear that the first approximation as a result of starting with x1=10 is 10.8 (10.75 expressed to the nearest tenth).
If we start with x1=11 we get: 11,10.7272...,10.72380586,10.72380529. So whatever starting value we use we end up with this number which, when squared, = 115.
So if the answer is multi choice, then 10.8 is the first iteration to the nearest tenth.
