If
\(x_0\) is any positive rational number and
\[\begin{equation*}
x_1 = \frac{x_0 + 2}{x_0 + 1}, \quad x_2 = \frac{x_1 + 2}{x_1 + 1}, \quad x_3 = \frac{x_2 + 2}{x_2 + 1},
\end{equation*}\]
prove that
- \(\sqrt{2}\) lies between \(x_0\) and \(x_1\), and between \(x_2\) and \(x_3\),
- \(|x_1^2 - 2| < |x_0^2 - 2|\),
- \(|x_2^2 - 2| < \frac{1}{9} |x_0^2 - 2|\).
By taking
\(x_0 = \frac{7}{5}\), or otherwise, show that
\[\begin{equation*}
\frac{41}{29} < \sqrt{2} < \frac{99}{70},
\end{equation*}\]
and, without using tables, show that each of \(\dfrac{41}{29}\) and \(\dfrac{99}{70}\) differ from \(\sqrt{2}\) by less than \(5 \times 10^{-4}\).
