Question

What is the numerical value of\[ x = \frac{1}{2-\frac{1}{2-\frac{1}{2-\ldots}}} ?\]

Answer 1

1

Explanation:

We are asked to find the value of
`x` such that:

`x=(1)/(2-(1)/(2-\cdots))`

Notice that we can rewrite this as:

`x=(1)/(2-x)`

`(x)/(1)=(1)/(2-x)`

Cross multiply:

`x(2-x)=1`

Expand:

`2x-x^2=1`

Subtract 1 on both sides:

`2x-x^2-1=0`

Reorder:

`-x^2+2x-1=0`

Multiply both sides by -1:

`x^2-2x+1=0`

Factor as a binomial square on left:

`(x-1)^2=0`

This implies
`x-1=0`.

Add 1 on both sides:

`x=1`