`(1 - 1/3) (1 - 1/4)(1 - 1/5)………(1 - 1/n)` equals:

Options

A. `1/n`
B. `2/n`
C. `3/n`
D. `4/n`

Answer

To solve the expression \((1 - \frac{1}{3})(1 - \frac{1}{4})(1 - \frac{1}{5}) \ldots (1 - \frac{1}{n})\), we will break it down step by step.

### Step 1: Rewrite Each Term
We start by rewriting each term in the product:
\[ 1 - \frac{1}{k} = \frac{k - 1}{k} \]
for \(k = 3, 4, 5, \ldots, n\).

### Step 2: Substitute the Terms
Now we can substitute this into our product:
\[ (1 - \frac{1}{3})(1 - \frac{1}{4})(1 - \frac{1}{5}) \ldots (1 - \frac{1}{n}) = \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{4}{5} \cdots \frac{n-1}{n} \]

### Step 3: Write the Complete Product
The complete product can be written as:
\[ \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{4}{5} \cdots \frac{n-1}{n} = \frac{2 \cdot 3 \cdot 4 \cdots (n-1)}{3 \cdot 4 \cdots n} \]

### Step 4: Cancel Common Terms
Notice that in the numerator and denominator, many terms will cancel out:
\[ = \frac{2}{n} \]
because all terms from \(3\) to \(n-1\) in the numerator and denominator cancel each other.

### Step 5: Final Result
Thus, the final result of the expression is:
\[ \frac{2}{n} \]