The value of

$1-\frac{1}{1+\frac{1}{2-\frac{1}{2+\frac{1}{3}}}}is:$

The correct answer is 7/18.

We are asked to find the value of the given continued fraction expression:
$1-\frac{1}{1+\frac{1}{2-\frac{1}{2+\frac{1}{3}}}}$

To evaluate this expression, we solve from the innermost denominator at the bottom step-by-step upward.

Step 1: Simplify the bottom-most addition:
$2+\frac{1}{3}=\frac{6+1}{3}=\frac{7}{3}$

Step 2: Substitute this result into the next denominator layer:
$2-\frac{1}{7/3}=2-\frac{3}{7}=\frac{14-3}{7}=\frac{11}{7}$

Step 3: Substitute this value into the next layer:
$1+\frac{1}{11/7}=1+\frac{7}{11}=\frac{11+7}{11}=\frac{18}{11}$

Step 4: Evaluate the fraction with this denominator:
$\frac{1}{18/11}=\frac{11}{18}$

Step 5: Perform the final subtraction:
$1-\frac{11}{18}=\frac{18-11}{18}=\frac{7}{18}$

Thus, the value of the given expression is 7/18.