Question

Given `x=sqrt(7+sqrt(7+sqrt(7+sqrt(7+cdots))))` and `y=sqrt(7-sqrt(7-sqrt(7-sqrt(7-cdots))))` calculate `x-y =` ?

Answer 1

`x-y=1`

Explanation:

From

`x=sqrt(7+sqrt(7+sqrt(7+sqrt(7+cdots))))`

and

`y=sqrt(7-sqrt(7-sqrt(7-sqrt(7-cdots))))`

we get

`x=sqrt(7+x)`
`y=sqrt(7-y)`

now squaring both sides

`x^2=7+x`
`y^2=7-y`

now subtracting side by side

`x^2-y^2=x+y rArr (x+y)(x-y)=x+y rArr x-y = 1`