# How do you prove that the 8-sd approximation to the value of the infinite continued fraction #0.0123456789/(1+0.0123456789/(1+0.0123456789/(1+...))) =0.012196914# ?

##### 2 Answers

Express as a solution of a quadratic, applying the quadratic formula to find value:

#(-1+sqrt(1.0493827156))/2~~0.012196914#

#### Explanation:

Let

We want to find the value of

#x = k/(1+k/(1+k/(1+...)))#

Then:

#k/x = k -: (k/(1+k/(1+k/(1+...)))) = 1+k/(1+k/(1+k/(1+...))) = 1+x#

Multiplying both ends by

#x+x^2 = k#

Subtract

#x^2+x-k = 0#

Using the quadratic formula:

#x = (-1+-sqrt(1+4k))/2#

Since

#x = (-1+sqrt(1+4k))/2#

#= (-1+sqrt(1+(4*0.0123456789)))/2#

#=(-1+sqrt(1.0493827156))/2#

#~~0.01219691418437889686#

#~~0.012196914# to#8# s.d.

See below

#### Explanation:

It is an infinite expansion

but

then

but

then