The value of `1 + (1)/( 1 + (1)/(1+(1)/(1+(2)/(3))))` is

Options

A. `(21)/(13)`
B. `(17)/(3)`
C. `(34)/(21)`
D. `(8)/(5)`

Answer

To solve the expression \(1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{2}{3}}}}\), we will work from the innermost part of the expression outward.

### Step-by-Step Solution:

1. **Start with the innermost fraction:**
\[ \frac{2}{3} \]
This is already simplified.

2. **Next, evaluate the expression:**
\[ 1 + \frac{2}{3} \]
To add these, convert 1 to a fraction:
\[ 1 = \frac{3}{3} \]
Now, add:
\[ \frac{3}{3} + \frac{2}{3} = \frac{5}{3} \]

3. **Now substitute this back into the next layer:**
\[ 1 + \frac{5}{3} \]
Again, convert 1 to a fraction:
\[ 1 = \frac{3}{3} \]
Now, add:
\[ \frac{3}{3} + \frac{5}{3} = \frac{8}{3} \]

4. **Substitute this result into the next layer:**
\[ 1 + \frac{8}{3} \]
Convert 1 to a fraction:
\[ 1 = \frac{3}{3} \]
Now, add:
\[ \frac{3}{3} + \frac{8}{3} = \frac{11}{3} \]

5. **Finally, substitute this into the outermost layer:**
\[ 1 + \frac{11}{3} \]
Convert 1 to a fraction:
\[ 1 = \frac{3}{3} \]
Now, add:
\[ \frac{3}{3} + \frac{11}{3} = \frac{14}{3} \]

### Final Answer:
The value of the expression \(1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{2}{3}}}}\) is \(\frac{14}{3}\).