If `f(x)={x}+{x+[(x)/(1+x^(2))]}+{x+[(x)/(1+2x^(2))]}+{x+[(x)/(1+3x^(2))]}.......+{x+[(x)/(1+99x^(2))]}`, then values of `[f(sqrt(3))]` is where `[*]` denotes greatest integer function and `{*}` represent fractional part function)

Options

A. `5050`
B. `4950`
C. `17`
D. `73`

Answer

To solve the problem, we need to evaluate the function \( f(x) \) defined as:

\[ f(x) = \sum_{k=0}^{99} \left( x + \frac{x}{1 + kx^2} \right) \]

We will substitute \( x = \sqrt{3} \) into the function and simplify it step by step.

### Step 1: Substitute \( x = \sqrt{3} \)

Substituting \( x = \sqrt{3} \) into the function gives:

\[ f(\sqrt{3}) = \sum_{k=0}^{99} \left( \sqrt{3} + \frac{\sqrt{3}}{1 + k(\sqrt{3})^2} \right) \]

### Step 2: Simplify the expression inside the summation

Since \( (\sqrt{3})^2 = 3 \), we can rewrite the term \( 1 + k(\sqrt{3})^2 \) as \( 1 + 3k \). Thus, we have:

\[ f(\sqrt{3}) = \sum_{k=0}^{99} \left( \sqrt{3} + \frac{\sqrt{3}}{1 + 3k} \right) \]

### Step 3: Break the summation into two parts

We can separate the summation into two distinct parts:

\[ f(\sqrt{3}) = \sum_{k=0}^{99} \sqrt{3} + \sum_{k=0}^{99} \frac{\sqrt{3}}{1 + 3k} \]

### Step 4: Calculate the first summation

The first summation is straightforward:

\[ \sum_{k=0}^{99} \sqrt{3} = 100\sqrt{3} \]

### Step 5: Calculate the second summation

Now we need to evaluate the second summation:

\[ \sum_{k=0}^{99} \frac{\sqrt{3}}{1 + 3k} \]

This can be approximated by recognizing that as \( k \) increases, the term \( 1 + 3k \) grows larger, making the fraction smaller.

### Step 6: Estimate the second summation

To estimate this summation, we can calculate the first few terms and observe the pattern. The first few terms are:

- For \( k=0 \): \( \frac{\sqrt{3}}{1} = \sqrt{3} \)
- For \( k=1 \): \( \frac{\sqrt{3}}{4} \)
- For \( k=2 \): \( \frac{\sqrt{3}}{7} \)
- For \( k=3 \): \( \frac{\sqrt{3}}{10} \)

As \( k \) increases, the terms decrease. We can approximate the sum using integrals or numerical methods, but for simplicity, we can note that the sum will be significantly smaller than the first part.

### Step 7: Combine the results

Let’s denote the second summation as \( S \):

\[ S = \sum_{k=0}^{99} \frac{\sqrt{3}}{1 + 3k} \]

The total function evaluates to:

\[ f(\sqrt{3}) \approx 100\sqrt{3} + S \]

### Step 8: Calculate the approximate value

Using \( \sqrt{3} \approx 1.732 \):

\[ 100\sqrt{3} \approx 173.2 \]

Assuming \( S \) is small, we can estimate \( f(\sqrt{3}) \approx 173.2 + S \).

### Step 9: Apply the greatest integer function

Now, we need to find \( [f(\sqrt{3})] \), where \( [*] \) denotes the greatest integer function. Since \( S \) is small, we can estimate:

\[ [f(\sqrt{3})] \approx [173.2 + S] \approx 173 \]

### Final Answer

Thus, the value of \( [f(\sqrt{3})] \) is:

\[ \boxed{173} \]