`[-(1)/(3)]+[-(1)/(3)-(1)/(100)]+[-(1)/(3)-(2)/(100)]+….+[-(1)/(3)-(99)/(100)]` is equal to (where `[.]` denotes greatest integer function)

Options

A. `-132`
B. `-133`
C. `-134`
D. `-131`

Answer

To solve the expression

\[ [-\frac{1}{3}] + [-\frac{1}{3} - \frac{1}{100}] + [-\frac{1}{3} - \frac{2}{100}] + \ldots + [-\frac{1}{3} - \frac{99}{100}] \]

we will break it down step by step.

### Step 1: Determine the value of \([- \frac{1}{3}]\)

The greatest integer function, denoted by \([.]\), gives the largest integer less than or equal to the number.

\[ -\frac{1}{3} \approx -0.333 \]

Thus,

\[ [-\frac{1}{3}] = -1 \]

**Hint:** The greatest integer function rounds down to the nearest whole number.

### Step 2: Analyze the general term

The general term in the series can be expressed as:

\[ [-\frac{1}{3} - \frac{k}{100}] \text{ for } k = 0, 1, 2, \ldots, 99 \]

This can be rewritten as:

\[ [-\frac{1}{3} - \frac{k}{100}] = [-\frac{1}{3} - \frac{k}{100}] = [-\frac{100 + k}{300}] \]

### Step 3: Determine the range of the terms

Now we need to evaluate the expression:

\[ -\frac{100 + k}{300} \]

As \(k\) varies from \(0\) to \(99\), the value of \(-\frac{100 + k}{300}\) will range from:

- For \(k = 0\):
\[ -\frac{100 + 0}{300} = -\frac{100}{300} = -\frac{1}{3} \]

- For \(k = 99\):
\[ -\frac{100 + 99}{300} = -\frac{199}{300} \]

### Step 4: Determine the greatest integer value for the range

The values of \(-\frac{100 + k}{300}\) will be between \(-\frac{1}{3}\) and \(-\frac{199}{300}\).

Since both of these values are between \(-1\) and \(0\), we can conclude that:

\[ [-\frac{100 + k}{300}] = -1 \text{ for } k = 0 \text{ to } 66 \]
\[ [-\frac{100 + k}{300}] = -2 \text{ for } k = 67 \text{ to } 99 \]

### Step 5: Count the number of terms

- For \(k = 0\) to \(66\), there are \(67\) terms where the greatest integer is \(-1\).
- For \(k = 67\) to \(99\), there are \(33\) terms where the greatest integer is \(-2\).

### Step 6: Calculate the total sum

The total sum can now be calculated as:

\[ 67 \times (-1) + 33 \times (-2) = -67 - 66 = -133 \]

### Final Answer

Thus, the value of the original expression is:

\[ \boxed{-133} \]