Find the value of the expression: 1+1/ (1+1/ (1+1/ (1+1 /x)))

Step-by-Step Solution

1. Solve the innermost part: 1 + 1/x = (x + 1) / x

2. Move to the next level: The expression becomes 1 + 1 / ((x + 1) / x). This simplifies to 1 + x / (x + 1). To add these, find a common denominator: (x + 1 + x) / (x + 1) = (2x + 1) / (x + 1)

3. Move to the final level: The full expression is 1 + 1 / ((2x + 1) / (x + 1)). This simplifies to 1 + (x + 1) / (2x + 1). Again, find a common denominator: (2x + 1 + x + 1) / (2x + 1) = (3x + 2) / (2x + 1)

Note on Option A: The expression provided in the option (5x + 3) / (3x + 2) corresponds to a continued fraction with one additional level of nesting: 1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / x))).