The value of `[(1)/(1xx2)+(1)/(2xx 3)+(1)/(3xx4)+"…………"+(1)/(49xx50)]` will be

Options

A. `(49)/(50)`
B. `(1)/(5000)`
C. `(51)/(50)`
D. `(101)/(500)`

Answer

To solve the given expression \(\frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4} + \ldots + \frac{1}{49 \times 50}\), we can use a technique involving partial fractions.

### Step-by-Step Solution:

1. **Rewrite Each Term**:
Each term in the series can be rewritten using the identity:
\[ \frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1} \]
Therefore, we can express the series as:
\[ \frac{1}{1 \times 2} = 1 - \frac{1}{2}, \quad \frac{1}{2 \times 3} = \frac{1}{2} - \frac{1}{3}, \quad \frac{1}{3 \times 4} = \frac{1}{3} - \frac{1}{4}, \ldots, \frac{1}{49 \times 50} = \frac{1}{49} - \frac{1}{50} \]

2. **Write the Full Series**:
The entire series can now be written as:
\[ \left(1 - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \left(\frac{1}{3} - \frac{1}{4}\right) + \ldots + \left(\frac{1}{49} - \frac{1}{50}\right) \]

3. **Observe the Cancellation**:
When we add these terms together, we notice that most terms will cancel out:
\[ 1 - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + \ldots - \frac{1}{50} \]
All intermediate terms cancel, leaving us with:
\[ 1 - \frac{1}{50} \]

4. **Simplify the Result**:
Now, we can simplify the remaining expression:
\[ 1 - \frac{1}{50} = \frac{50}{50} - \frac{1}{50} = \frac{49}{50} \]

### Final Answer:
The value of the given series is:
\[ \frac{49}{50} \]