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:
 * $\sqrt {29} = \sqbrk {5, \sequence {2, 1, 1, 2, 10} }$

Proof
Let $\sqrt {29} = \sqbrk {a_0, a_1, a_2, a_3, \ldots}$

From Partial Quotients of Continued Fraction Expansion of Irrational Square Root, the partial quotients of this continued fraction expansion can be calculated as:


 * $a_r = \floor {\dfrac {\floor {\sqrt {29} } + P_r} {Q_r} }$

where:


 * $P_r = \begin {cases} 0 & : r = 0 \\

a_{r - 1} Q_{r - 1} - P_{r - 1} & : r > 0 \\ \end {cases}$


 * $Q_r = \begin {cases} 1 & : r = 0 \\

\dfrac {n - {P_r}^2} {Q_{r - 1} } & : r > 0 \\ \end {cases}$


 * }

and the cycle is complete:
 * $\sequence {2, 1, 1, 2, 10}$