`1+1/(1+1/(1+1/4))=?`

Options

A. `14/9`
B. `2 5/9`
C. `3 5/9`
D. `4 5/9`

Answer

To solve the expression \( 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{4}}} \), we will work from the innermost fraction outward.

### Step-by-step Solution:

1. **Start with the innermost fraction**:
\[ 1 + \frac{1}{4} \]
To solve this, we convert it to a single fraction:
\[ = \frac{4}{4} + \frac{1}{4} = \frac{4 + 1}{4} = \frac{5}{4} \]

2. **Replace the innermost part in the original expression**:
Now our expression becomes:
\[ 1 + \frac{1}{1 + \frac{5}{4}} \]

3. **Solve the next fraction**:
We need to solve:
\[ 1 + \frac{5}{4} \]
Again, convert it to a single fraction:
\[ = \frac{4}{4} + \frac{5}{4} = \frac{4 + 5}{4} = \frac{9}{4} \]

4. **Replace this back into the expression**:
Now our expression is:
\[ 1 + \frac{1}{\frac{9}{4}} \]

5. **Simplify the fraction**:
The fraction \( \frac{1}{\frac{9}{4}} \) can be simplified:
\[ = \frac{4}{9} \]

6. **Replace this back into the expression**:
Now we have:
\[ 1 + \frac{4}{9} \]

7. **Combine the final fractions**:
Convert \( 1 \) into a fraction with a denominator of \( 9 \):
\[ = \frac{9}{9} + \frac{4}{9} = \frac{9 + 4}{9} = \frac{13}{9} \]

8. **Final Answer**:
Thus, the value of the expression \( 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{4}}} \) is:
\[ \frac{13}{9} \]