# Continued Fraction Expansion of Irrational Square Root/Examples/61/Convergents

## Convergents to Continued Fraction Expansion of $\sqrt {61}$

The sequence of convergents to the continued fraction expansion of the square root of $61$ begins:

$\dfrac 7 1, \dfrac {8} 1, \dfrac {39} 5, \dfrac {125} {16}, \dfrac {164} {21}, \dfrac {453} {58}, \dfrac {1070} {137}, \dfrac {1523} {195}, \dfrac {5639} {722}, \dfrac {24079} {3083}, \ldots$

## Proof

Let $\sqbrk {a_0, a_1, a_2, \ldots}$ be its continued fraction expansion.

Let $\sequence {p_n}_{n \ge \mathop 0}$ and $\sequence {q_n}_{n \ge \mathop 0}$ be its numerators and denominators.

Then the $n$th convergent is $p_n / q_n$.

By definition:

$p_k = \begin {cases} a_0 & : k = 0 \\ a_0 a_1 + 1 & : k = 1 \\ a_k p_{k - 1} + p_{k - 2} & : k > 1 \end {cases}$
$q_k = \begin {cases} 1 & : k = 0 \\ a_1 & : k = 1 \\ a_k q_{k - 1} + q_{k - 2} & : k > 1 \end {cases}$
$\sqrt {61} = \sqbrk {7, \sequence {1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14} }$

Thus the convergents are assembled:

$k$ $a_k$ $p_k = a_k p_{k - 1} + p_{k - 2}$ $q_k = a_k q_{k - 1} + q_{k - 2}$ $\dfrac {p_k} {q_k}$ Decimal value
$0$ $7$ $7$ $1$ $\dfrac { 7 } 1$ $7$
$1$ $1$ $7 \times 1 + 1 = 8$ $1$ $\dfrac { 8 } { 1 }$ $8$
$2$ $4$ $4 \times 8 + 7 = 39$ $4 \times 1 + 1 = 5$ $\dfrac { 39 } { 5 }$ $7.8$
$3$ $3$ $3 \times 39 + 8 = 125$ $3 \times 5 + 1 = 16$ $\dfrac { 125 } { 16 }$ $7.8125$
$4$ $1$ $1 \times 125 + 39 = 164$ $1 \times 16 + 5 = 21$ $\dfrac { 164 } { 21 }$ $7.8095238095$
$5$ $2$ $2 \times 164 + 125 = 453$ $2 \times 21 + 16 = 58$ $\dfrac { 453 } { 58 }$ $7.8103448276$
$6$ $2$ $2 \times 453 + 164 = 1070$ $2 \times 58 + 21 = 137$ $\dfrac { 1070 } { 137 }$ $7.8102189781$
$7$ $1$ $1 \times 1070 + 453 = 1523$ $1 \times 137 + 58 = 195$ $\dfrac { 1523 } { 195 }$ $7.8102564103$
$8$ $3$ $3 \times 1523 + 1070 = 5639$ $3 \times 195 + 137 = 722$ $\dfrac { 5639 } { 722 }$ $7.8102493075$
$9$ $4$ $4 \times 5639 + 1523 = 24079$ $4 \times 722 + 195 = 3083$ $\dfrac { 24079 } { 3083 }$ $7.8102497567$
$10$ $1$ $1 \times 24079 + 5639 = 29718$ $1 \times 3083 + 722 = 3805$ $\dfrac { 29718 } { 3805 }$ $7.8102496715$
$11$ $14$ $14 \times 29718 + 24079 = 440131$ $14 \times 3805 + 3083 = 56353$ $\dfrac { 440131 } { 56353 }$ $7.8102496761$
$12$ $1$ $1 \times 440131 + 29718 = 469849$ $1 \times 56353 + 3805 = 60158$ $\dfrac { 469849 } { 60158 }$ $7.8102496759$
$13$ $4$ $4 \times 469849 + 440131 = 2319527$ $4 \times 60158 + 56353 = 296985$ $\dfrac { 2319527 } { 296985 }$ $7.8102496759$
$14$ $3$ $3 \times 2319527 + 469849 = 7428430$ $3 \times 296985 + 60158 = 951113$ $\dfrac { 7428430 } { 951113 }$ $7.8102496759$
$15$ $1$ $1 \times 7428430 + 2319527 = 9747957$ $1 \times 951113 + 296985 = 1248098$ $\dfrac { 9747957 } { 1248098 }$ $7.8102496759$
$16$ $2$ $2 \times 9747957 + 7428430 = 26924344$ $2 \times 1248098 + 951113 = 3447309$ $\dfrac { 26924344 } { 3447309 }$ $7.8102496759$
$17$ $2$ $2 \times 26924344 + 9747957 = 63596645$ $2 \times 3447309 + 1248098 = 8142716$ $\dfrac { 63596645 } { 8142716 }$ $7.8102496759$
$18$ $1$ $1 \times 63596645 + 26924344 = 90520989$ $1 \times 8142716 + 3447309 = 11590025$ $\dfrac { 90520989 } { 11590025 }$ $7.8102496759$
$19$ $3$ $3 \times 90520989 + 63596645 = 335159612$ $3 \times 11590025 + 8142716 = 42912791$ $\dfrac { 335159612 } { 42912791 }$ $7.8102496759$
$20$ $4$ $4 \times 335159612 + 90520989 = 1431159437$ $4 \times 42912791 + 11590025 = 183241189$ $\dfrac { 1431159437 } { 183241189 }$ $7.8102496759$
$21$ $1$ $1 \times 1431159437 + 335159612 = 1766319049$ $1 \times 183241189 + 42912791 = 226153980$ $\dfrac { 1766319049 } { 226153980 }$ $7.8102496759$

$\blacksquare$