Review question

# If $x_1 = (x_0 + 2)/(x_0 + 1)$, can we show $\sqrt{2}$ lies between $x_0$ and $x_1$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8404

## Question

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

1. $\sqrt{2}$ lies between $x_0$ and $x_1$, and between $x_2$ and $x_3$,
2. $|x_1^2 - 2| < |x_0^2 - 2|$,
3. $|x_2^2 - 2| < \frac{1}{9} |x_0^2 - 2|$.
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 $5 \times 10^{-4}$.