Question

The infinite continued fraction representation of √5 is [2,4,4,4,4,...]. Give the first four decimal estimates of √5 coming from the partial convergents of this continued fraction. (Round each to 6 decimal places.)

Answer 1

Given, infinite continued fraction representation of sqrt(5) is [2,4,4,4,4,...], i.e., a0=2, a1=4, a2=4, a3=4, a4=4

The partial convergents of our continued fraction repr. Of sqrt(5) are, say: P_{​​​​​​0}​​​​​, P_{​​​​​​1}, P_​2, P_{​​​​​​3}, P₄

P_{​​​​0}=a0 = 2

P_{​​​​​​1}= = 2+(1/4) = 2.25

P_​2= = 2+(1/(4+(1/4))) = 2+(1/4.25) = 2+0.235294118 = 2.235294118 = 2.235294 (Round to 6 decimal places)

P_{​​​​​​3}= =2+(1/(4+(1/4+(1/4))))=2+(1/(4+0.235294118)) = 2+0.236111111 = 2.236111111 = 2.236111 (Round to 6 decimal places)

P_{​​​​​​4} = = 2+(1/(4+(1/4+(1/(4+(1/4))))))

= 2+(1/(4+0.236111111)) = 2+0.236065574 = 2.236065574 = 2.236066 (Round to 6 decimal places)

Similarly,

P_{​​​​​​5}= 2.236068111 = 2.236068 (Round to 6 decimal places)

P_{​​​​​​6}= 2.23606797 = 2.236068 (Round to 6 decimal places)

Hence, approximation of = 2.236068