The value of
`1000[(1)/(1xx2)+(1)/(2xx3)+(1)/(3xx4)+cdots+(1)/(999xx1000)]` is equal to

Options

A. 1000
B. 999
C. 1001
D. `1//999`

Answer

To solve the problem, we need to evaluate the expression:

\[ 1000\left[\frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4} + \cdots + \frac{1}{999 \times 1000}\right] \]

### Step 1: Rewrite each term in the series

Each term in the series can be rewritten using the formula:

\[ \frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1} \]

Thus, we can rewrite the series as:

\[ \frac{1}{1 \times 2} = 1 - \frac{1}{2}, \quad \frac{1}{2 \times 3} = \frac{1}{2} - \frac{1}{3}, \quad \frac{1}{3 \times 4} = \frac{1}{3} - \frac{1}{4}, \ldots, \quad \frac{1}{999 \times 1000} = \frac{1}{999} - \frac{1}{1000} \]

### Step 2: Write the entire series

Now, substituting back into the series, we have:

\[ \sum_{n=1}^{999} \left(\frac{1}{n} - \frac{1}{n+1}\right) \]

This is a telescoping series. When we write out the first few and the last few terms, we see that most terms cancel out:

\[ \left(1 - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \left(\frac{1}{3} - \frac{1}{4}\right) + \cdots + \left(\frac{1}{999} - \frac{1}{1000}\right) \]

### Step 3: Simplify the series

After cancellation, we are left with:

\[ 1 - \frac{1}{1000} \]

### Step 4: Calculate the value of the series

So, the value of the series is:

\[ 1 - \frac{1}{1000} = \frac{1000 - 1}{1000} = \frac{999}{1000} \]

### Step 5: Multiply by 1000

Now, we multiply this result by 1000:

\[ 1000 \times \frac{999}{1000} = 999 \]

### Final Answer

Thus, the value of the original expression is:

\[ \boxed{999} \]

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