If `x = 1 + (1)/(1 + (1)/(1 + (1)/(1 + (1)/(2))))` then, the value of `2x + (7)/(4)` is :

Options

A. 3
B. 4
C. 5
D. 6

Answer

To solve the expression \( x = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{2}}}} \) and find the value of \( 2x + \frac{7}{4} \), we will simplify it step by step.

### Step 1: Simplify the innermost fraction
Start from the innermost part of the expression:
\[ \frac{1}{2} = 0.5 \]
Now, we substitute this back into the expression:
\[ x = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{0.5}}} \]

### Step 2: Simplify the next fraction
Now, simplify \( 1 + \frac{1}{0.5} \):
\[ \frac{1}{0.5} = 2 \quad \text{so} \quad 1 + 2 = 3 \]
Now, substitute this back into the expression:
\[ x = 1 + \frac{1}{1 + \frac{1}{3}} \]

### Step 3: Simplify the next fraction
Now, simplify \( 1 + \frac{1}{3} \):
\[ \frac{1}{3} = 0.333 \quad \text{so} \quad 1 + 0.333 = \frac{4}{3} \]
Now, substitute this back into the expression:
\[ x = 1 + \frac{1}{\frac{4}{3}} = 1 + \frac{3}{4} \]

### Step 4: Simplify the next fraction
Now, simplify \( 1 + \frac{3}{4} \):
\[ 1 = \frac{4}{4} \quad \text{so} \quad \frac{4}{4} + \frac{3}{4} = \frac{7}{4} \]
Now, substitute this back into the expression:
\[ x = \frac{7}{4} \]

### Step 5: Calculate \( 2x + \frac{7}{4} \)
Now, we need to calculate \( 2x + \frac{7}{4} \):
\[ 2x = 2 \times \frac{7}{4} = \frac{14}{4} = \frac{7}{2} \]
Now add \( \frac{7}{4} \):
\[ 2x + \frac{7}{4} = \frac{7}{2} + \frac{7}{4} \]
To add these fractions, we need a common denominator. The common denominator of 2 and 4 is 4:
\[ \frac{7}{2} = \frac{14}{4} \]
Now, add:
\[ \frac{14}{4} + \frac{7}{4} = \frac{21}{4} \]

### Final Answer
Thus, the value of \( 2x + \frac{7}{4} \) is:
\[ \frac{21}{4} \]