`999(1)/(7)+999(2)/(7)+999(3)/(7)+999(4)/(7)+999(5)/(7)+999(6)/(7)` is simplified to:

Options

A. 5997
B. 5979
C. 5994
D. 2997

Answer

To simplify the expression \( 999\left(\frac{1}{7}\right) + 999\left(\frac{2}{7}\right) + 999\left(\frac{3}{7}\right) + 999\left(\frac{4}{7}\right) + 999\left(\frac{5}{7}\right) + 999\left(\frac{6}{7}\right) \), we can follow these steps:

### Step 1: Rewrite the Expression
We can rewrite each term in the expression:
\[ 999\left(\frac{1}{7}\right) = 999 + \frac{1}{7} \]
\[ 999\left(\frac{2}{7}\right) = 999 + \frac{2}{7} \]
\[ 999\left(\frac{3}{7}\right) = 999 + \frac{3}{7} \]
\[ 999\left(\frac{4}{7}\right) = 999 + \frac{4}{7} \]
\[ 999\left(\frac{5}{7}\right) = 999 + \frac{5}{7} \]
\[ 999\left(\frac{6}{7}\right) = 999 + \frac{6}{7} \]

### Step 2: Combine Like Terms
Now, we can combine all the \(999\) terms:
\[ 999 + 999 + 999 + 999 + 999 + 999 = 999 \times 6 = 5994 \]

### Step 3: Combine the Fractions
Next, we combine the fractional parts:
\[ \frac{1}{7} + \frac{2}{7} + \frac{3}{7} + \frac{4}{7} + \frac{5}{7} + \frac{6}{7} = \frac{1 + 2 + 3 + 4 + 5 + 6}{7} \]
Calculating the sum of the numerators:
\[ 1 + 2 + 3 + 4 + 5 + 6 = 21 \]
Thus, we have:
\[ \frac{21}{7} = 3 \]

### Step 4: Add the Results
Now we can add the results from Step 2 and Step 3:
\[ 5994 + 3 = 5997 \]

### Final Result
The simplified value of the given expression is:
\[ \boxed{5997} \]