14. Determine the value of: \( 1+\frac{1}{1+\frac{1}{11+\frac{1}{8}}} \) \( \begin{array}{llll}\text { (A) } \frac{26}{12} & \text { (B) } \frac{25}{17} & \text { (C) } \frac{24}{17} & \text { (D) } \frac{23}{17}\end{array} \)

To evaluate the expression \( 1+\frac{1}{1+\frac{1}{11+\frac{1}{8}}} \), we can break it down step by step. 1. Start with the innermost expression: \[ 11 + \frac{1}{8} \] This can be rewritten as: \[ 11 + 0.125 = 11.125 = \frac{89}{8} \] (since \( 11 = \frac{88}{8} \) and \( \frac{88}{8} + \frac{1}{8} = \frac{89}{8} \)). 2. Next, substitute this back into the previous fraction: \[ 1 + \frac{1}{\frac{89}{8}} = 1 + \frac{8}{89} \] To combine these, convert \( 1 \) into a fraction: \[ 1 = \frac{89}{89} \implies 1 + \frac{8}{89} = \frac{89 + 8}{89} = \frac{97}{89} \] 3. Now we substitute this back into the expression: \[ \frac{1}{\frac{97}{89}} = \frac{89}{97} \] 4. Finally, we add 1 to this: \[ 1 + \frac{89}{97} = \frac{97}{97} + \frac{89}{97} = \frac{186}{97} \] Now we can express this final sum: \[ 1 + \frac{1}{1+\frac{1}{11+\frac{1}{8}}} = \frac{186}{97} \] However, this is not one of the provided answer choices. Let's double-check for any potential errors or simplifications. After recalculating, it seems our earlier math holds up with no discrepancies found. Calculation steps are aligned correctly. The options given were not reflective of the computed result. Hence, the value of the given expression \( 1+\frac{1}{1+\frac{1}{11+\frac{1}{8}}} \) evaluates to \( \frac{186}{97} \), which highlights either an issue with the options or in the original question setup itself. Upon review, the provided answer choices may not correspond accurately to the calculated result. But sticking to your options, the correct breakdown we provided does not lead to any of the given choices. Always be cautious of potential transcription errors in such problems!