Simplify the following expressions:
`1 + 1/(1+1/(1+1/(1+1/(1+1/(1)))))`

Answer

To simplify the expression \( 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1}}}}} \), we will work from the innermost fraction outward. Let's break it down step by step.

### Step 1: Start from the innermost fraction
The innermost part of the expression is \( 1 \). So we can write:
\[ 1 + 1 = 2 \]

### Step 2: Substitute back into the expression
Now we replace the innermost part with \( 2 \):
\[ 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + 2}}} \]

### Step 3: Simplify the next fraction
Now we simplify \( 1 + 2 \):
\[ 1 + 2 = 3 \]
So we have:
\[ 1 + \frac{1}{1 + \frac{1}{1 + 3}} \]

### Step 4: Simplify the next fraction
Now we simplify \( 1 + 3 \):
\[ 1 + 3 = 4 \]
So we have:
\[ 1 + \frac{1}{1 + \frac{1}{4}} \]

### Step 5: Simplify the next fraction
Now we simplify \( 1 + \frac{1}{4} \):
To do this, we find a common denominator:
\[ 1 = \frac{4}{4} \implies 1 + \frac{1}{4} = \frac{4}{4} + \frac{1}{4} = \frac{5}{4} \]
So we have:
\[ 1 + \frac{1}{\frac{5}{4}} \]

### Step 6: Simplify the next fraction
Now we simplify \( \frac{1}{\frac{5}{4}} \):
This is equivalent to:
\[ \frac{4}{5} \]
So we have:
\[ 1 + \frac{4}{5} \]

### Step 7: Combine the last fractions
Now we combine \( 1 + \frac{4}{5} \):
To do this, we find a common denominator:
\[ 1 = \frac{5}{5} \implies 1 + \frac{4}{5} = \frac{5}{5} + \frac{4}{5} = \frac{9}{5} \]

### Final Result
Thus, the simplified expression is:
\[ \frac{9}{5} \]