The expression `(1+1/3)(1+1/4)(1+1/5)…(1+1/n)` simplifies to :

Options

A. `(n+1)/3`
B. `n/(n+1)`
C. `3/n`
D. `1+1/3, 1/4, 1/5 … 1/n`

Answer

To simplify the expression \((1 + \frac{1}{3})(1 + \frac{1}{4})(1 + \frac{1}{5}) \ldots (1 + \frac{1}{n})\), we can follow these steps:

### Step 1: Rewrite Each Term
Each term in the expression can be rewritten as follows:
\[ 1 + \frac{1}{k} = \frac{k + 1}{k} \]
for \(k = 3, 4, 5, \ldots, n\).

### Step 2: Substitute and Expand
Substituting this into the expression gives:
\[ (1 + \frac{1}{3})(1 + \frac{1}{4})(1 + \frac{1}{5}) \ldots (1 + \frac{1}{n}) = \frac{4}{3} \cdot \frac{5}{4} \cdot \frac{6}{5} \cdots \frac{n + 1}{n} \]

### Step 3: Simplify the Product
Notice that in this product, most terms will cancel out:
\[ \frac{4}{3} \cdot \frac{5}{4} \cdot \frac{6}{5} \cdots \frac{n + 1}{n} = \frac{n + 1}{3} \]
Here, all intermediate terms cancel out, leaving us with:
\[ \frac{n + 1}{3} \]

### Step 4: Final Answer
Thus, the simplified expression is:
\[ \frac{n + 1}{3} \]