Evaluate $\sqrt{41}$ to $n$ decimal places without using a calculator. Evaluate $\sqrt{41}$ to $n$ decimal places without using a calculator.
$$x^2 -10x-16=0$$
I was asked to solve the above quadratic giving the solutions to one decimal place.
Using the quadratic formula I got: $x= 5\pm \sqrt{41}$ 
Using a calculator that is: $11.4(1dp)$ or $-1.4(1dp)$
Then I wondered if I could solve this without a calculator.
It can be done by trial and error squaring values and gradually getting closer to the required accuracy. Here are the values that I squared to get to the required accuracy for the above case. $6.5,6.4,6.45,6.43,6.41$ and $6.405$. But is there another cleaner way of solving a surd to some number of decimal places?

Hint. Construct a recurrence sequence $x_n=f(x_{n-1})$ for $n\geq 1$, with
$$f(x)=\frac{1}{2}\left(x+\frac{41}{x}\right).$$
A reasonable starting point would be $x_0=7$ (note that $6^2<41<7^2$).
For details take a look here.
