The value of (1-$\frac{1}{2}$)(1-$\frac{1}{3})$(1-$\frac{1}{4}$) .......(1-$\frac{1}{n}$) is :

Q: The value of (1-$\frac{1}{2}$)(1-$\frac{1}{3})$(1-$\frac{1}{4}$) .......(1-$\frac{1}{n}$) is :

To determine the value of the expression $(1-\frac{1}{2})(1-\frac{1}{3})(1-\frac{1}{4}) \ldots (1-\frac{1}{n})$, we need to simplify it step by step. ### Explanation for Option 1: 1. **Simplification of Each Term:** Each term in the product can be rewritten as: \[ 1 - \frac{1}{k} = \frac{k-1}{k} \] Therefore, the entire expression becomes: \[ \left(\frac{1}{2}\right)\left(\frac{2}{3}\right)\left(\frac{3}{4}\right) \ldots \left(\frac{n-1}{n}\right) \] 2. **Cancellation of Terms:** Notice that in this product, each numerator cancels with the denominator of the subsequent fraction: \[ \frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} \times \ldots \times \frac{n-1}{n} = \frac{1 \cdot 2 \cdot 3 \cdot \ldots \cdot (n-1)}{2 \cdot 3 \cdot 4 \cdot \ldots \cdot n} \] This simplifies to: \[ \frac{1}{n} \] 3. **Conclusion:** The entire product simplifies to $\frac{1}{n}$, which matches Option 1. ### Explanation for Why Other Options Are Incorrect: - **Option 2: $\frac{2}{n}$** - This option suggests that the product results in $\frac{2}{n}$. However, as shown in the simplification, the numerator is 1, not 2. Therefore, this option is incorrect. - **Option 3: $\frac{2(n-1)}{n}$** - This option implies a more complex numerator, $2(n-1)$, which does not arise from the cancellation process. The correct simplification results in a numerator of 1, making this option incorrect. - **Option 4: $\frac{2}{n(n+1)}$** - This option suggests a denominator of $n(n+1)$, which does not match the pattern of cancellation observed in the product. The correct denominator is simply $n$, not $n(n+1)$, thus making this option incorrect. In conclusion, the correct simplification of the given expression results in $\frac{1}{n}$, which is why Option 1 is the correct answer.

Question

Accepted Answer

To determine the value of the expression $(1-\frac{1}{2})(1-\frac{1}{3})(1-\frac{1}{4}) \ldots (1-\frac{1}{n})$, we need to simplify it step by step.

### Explanation for Option 1:

1. **Simplification of Each Term:**
   Each term in the product can be rewritten as:
   $$
   1 - \frac{1}{k} = \frac{k-1}{k}
   $$
   Therefore, the entire expression becomes:
   $$
   \left(\frac{1}{2}\right)\left(\frac{2}{3}\right)\left(\frac{3}{4}\right) \ldots \left(\frac{n-1}{n}\right)
   $$

2. **Cancellation of Terms:**
   Notice that in this product, each numerator cancels with the denominator of the subsequent fraction:
   $$
   \frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} \times \ldots \times \frac{n-1}{n} = \frac{1 \cdot 2 \cdot 3 \cdot \ldots \cdot (n-1)}{2 \cdot 3 \cdot 4 \cdot \ldots \cdot n}
   $$
   This simplifies to:
   $$
   \frac{1}{n}
   $$

3. **Conclusion:**
   The entire product simplifies to $\frac{1}{n}$, which matches Option 1.

### Explanation for Why Other Options Are Incorrect:

- **Option 2: $\frac{2}{n}$**
  - This option suggests that the product results in $\frac{2}{n}$. However, as shown in the simplification, the numerator is 1, not 2. Therefore, this option is incorrect.

- **Option 3: $\frac{2(n-1)}{n}$**
  - This option implies a more complex numerator, $2(n-1)$, which does not arise from the cancellation process. The correct simplification results in a numerator of 1, making this option incorrect.

- **Option 4: $\frac{2}{n(n+1)}$**
  - This option suggests a denominator of $n(n+1)$, which does not match the pattern of cancellation observed in the product. The correct denominator is simply $n$, not $n(n+1)$, thus making this option incorrect.

In conclusion, the correct simplification of the given expression results in $\frac{1}{n}$, which is why Option 1 is the correct answer.