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

1 Answer

2 votes

Answer:

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

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