Continued Fraction Expansion of Irrational Square Root/Examples/29

Examples of Continued Fraction Expansion of Irrational Square Root
The continued fraction expansion of the square root of $29$ is given by:
 * $\sqrt {29} = \left[{5, \left \langle{2, 1, 1, 2, 10}\right \rangle}\right]$

Proof
Let $\sqrt {29} = \left[{a_0, a_1, a_2, a_3, \ldots}\right]$

First note that:
 * $5^2 < 29 < \left({5 + 1}\right)^2$

and so $a_0 = 5$.

$a_1$:

$a_2$:

$a_3$:

$a_4$:

$a_5$:

and the cycle is complete:
 * $\left\langle{2, 1, 1, 2, 10}\right\rangle$