The value of `(1+1/x)(1+1/(x+1))(1+1/(x+2))(1+1/(x+3))` is equal to \

Options

A. `1+1/(x+4)`
B. `x+4`
C. `1/x`
D. `(x+4)/x`

Answer

To solve the expression \((1+1/x)(1+1/(x+1))(1+1/(x+2))(1+1/(x+3))\), we can simplify each term step by step.

### Step-by-Step Solution:

1. **Expand Each Term**:
Start by rewriting each term in the expression:
\[ (1 + \frac{1}{x}) = \frac{x + 1}{x} \]
\[ (1 + \frac{1}{x+1}) = \frac{x + 2}{x + 1} \]
\[ (1 + \frac{1}{x+2}) = \frac{x + 3}{x + 2} \]
\[ (1 + \frac{1}{x+3}) = \frac{x + 4}{x + 3} \]

2. **Combine the Terms**:
Now, substitute these back into the expression:
\[ \frac{x + 1}{x} \cdot \frac{x + 2}{x + 1} \cdot \frac{x + 3}{x + 2} \cdot \frac{x + 4}{x + 3} \]

3. **Cancel Common Factors**:
Notice that in the product, each numerator cancels with the denominator of the next term:
- \(x + 1\) in the numerator of the first term cancels with \(x + 1\) in the denominator of the second term.
- \(x + 2\) in the numerator of the second term cancels with \(x + 2\) in the denominator of the third term.
- \(x + 3\) in the numerator of the third term cancels with \(x + 3\) in the denominator of the fourth term.

After cancellation, we are left with:
\[ \frac{x + 4}{x} \]

4. **Final Result**:
Thus, the value of the original expression simplifies to:
\[ \frac{x + 4}{x} \]

### Final Answer:
The value of \((1+1/x)(1+1/(x+1))(1+1/(x+2))(1+1/(x+3))\) is \(\frac{x + 4}{x}\).