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The value of (1)/(2+(1)/(2-(1)/(2-(1)/(2...

The value of `(1)/(2+(1)/(2-(1)/(2-(1)/(2))))`

A

`(3)/(8) `

B

`(19)/(8)`

C

`(8)/(3)`

D

`(8)/(19)`

Text Solution

AI Generated Solution

The correct Answer is:
To simplify the expression \( \frac{1}{2 + \frac{1}{2 - \frac{1}{2 - \frac{1}{2}}}} \), we will work from the innermost fraction outwards. ### Step 1: Simplify the innermost fraction Start with the innermost part of the expression: \[ \frac{1}{2} \] This is simply \( 0.5 \). ### Step 2: Substitute into the next layer Now substitute \( \frac{1}{2} \) into the next layer: \[ 2 - \frac{1}{2} = 2 - 0.5 = 1.5 \] So, we have: \[ \frac{1}{2 - \frac{1}{2}} = \frac{1}{1.5} = \frac{2}{3} \] ### Step 3: Substitute into the next layer Now substitute \( \frac{2}{3} \) into the next layer: \[ 2 + \frac{2}{3} = \frac{6}{3} + \frac{2}{3} = \frac{8}{3} \] ### Step 4: Substitute into the outermost layer Finally, substitute \( \frac{8}{3} \) into the outermost fraction: \[ \frac{1}{\frac{8}{3}} = \frac{3}{8} \] ### Final Answer Thus, the value of the expression \( \frac{1}{2 + \frac{1}{2 - \frac{1}{2 - \frac{1}{2}}}} \) is: \[ \frac{3}{8} \] ---
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