`[(4)/(5)] + [(4)/(5)+(1)/(1000)] + [(4)/(5)+(2)/(1000)] + ...+[(4)/(5) + (999)/(1000)] =`
where [.] denotes greatest integer function

Options

A. 998
B. 980
C. 800
D. 801

Answer

To solve the given problem, we need to evaluate the sum of the series:

\[ \left\lfloor \frac{4}{5} \right\rfloor + \left\lfloor \frac{4}{5} + \frac{1}{1000} \right\rfloor + \left\lfloor \frac{4}{5} + \frac{2}{1000} \right\rfloor + \ldots + \left\lfloor \frac{4}{5} + \frac{999}{1000} \right\rfloor \]

### Step 1: Understanding the Greatest Integer Function
The greatest integer function, denoted as \(\lfloor x \rfloor\), gives the largest integer less than or equal to \(x\).

### Step 2: Calculate the First Term
The first term is:
\[ \left\lfloor \frac{4}{5} \right\rfloor = \left\lfloor 0.8 \right\rfloor = 0 \]

### Step 3: Calculate the General Term
The general term can be expressed as:
\[ \left\lfloor \frac{4}{5} + \frac{k}{1000} \right\rfloor \quad \text{for } k = 0, 1, 2, \ldots, 999 \]
This can be rewritten as:
\[ \left\lfloor 0.8 + \frac{k}{1000} \right\rfloor \]

### Step 4: Determine the Values of \(k\)
We need to determine when \(0.8 + \frac{k}{1000} \geq 1\):
\[ 0.8 + \frac{k}{1000} \geq 1 \implies \frac{k}{1000} \geq 0.2 \implies k \geq 200 \]
Thus, for \(k = 0\) to \(k = 199\), we have:
\[ \left\lfloor 0.8 + \frac{k}{1000} \right\rfloor = 0 \]
And for \(k = 200\) to \(k = 999\), we have:
\[ \left\lfloor 0.8 + \frac{k}{1000} \right\rfloor = 1 \]

### Step 5: Count the Terms
- From \(k = 0\) to \(k = 199\) (200 terms), the value is \(0\).
- From \(k = 200\) to \(k = 999\) (800 terms), the value is \(1\).

### Step 6: Calculate the Total Sum
The total sum can be calculated as:
\[ \text{Sum} = 0 \times 200 + 1 \times 800 = 800 \]

### Final Answer
Thus, the value of the given series is:
\[ \boxed{800} \]