If x = - 1 , then the value of ` (1)/( x^(99)) + (1)/( x^(98)) + (1)/( x^(97)) + (1)/( x^(96)) + (1)/( x^(95)) + (1)/( x^(94)) + (1)/( x) - 1 ` is

Options

A. 1
B. 0
C. `-2`
D. `-1`

Answer

To solve the expression \( \frac{1}{x^{99}} + \frac{1}{x^{98}} + \frac{1}{x^{97}} + \frac{1}{x^{96}} + \frac{1}{x^{95}} + \frac{1}{x^{94}} + \frac{1}{x} - 1 \) when \( x = -1 \), we will substitute \( x \) with \(-1\) and simplify step by step.

### Step 1: Substitute \( x = -1 \) into the expression

\[ \frac{1}{(-1)^{99}} + \frac{1}{(-1)^{98}} + \frac{1}{(-1)^{97}} + \frac{1}{(-1)^{96}} + \frac{1}{(-1)^{95}} + \frac{1}{(-1)^{94}} + \frac{1}{(-1)} - 1 \]

### Step 2: Calculate each term

- \( (-1)^{99} = -1 \) so \( \frac{1}{(-1)^{99}} = \frac{1}{-1} = -1 \)
- \( (-1)^{98} = 1 \) so \( \frac{1}{(-1)^{98}} = \frac{1}{1} = 1 \)
- \( (-1)^{97} = -1 \) so \( \frac{1}{(-1)^{97}} = \frac{1}{-1} = -1 \)
- \( (-1)^{96} = 1 \) so \( \frac{1}{(-1)^{96}} = \frac{1}{1} = 1 \)
- \( (-1)^{95} = -1 \) so \( \frac{1}{(-1)^{95}} = \frac{1}{-1} = -1 \)
- \( (-1)^{94} = 1 \) so \( \frac{1}{(-1)^{94}} = \frac{1}{1} = 1 \)
- \( \frac{1}{(-1)} = -1 \)

### Step 3: Combine all the terms

Now we can combine all the calculated values:

\[ -1 + 1 - 1 + 1 - 1 + 1 - 1 - 1 \]

### Step 4: Simplify the expression

Let's simplify this step by step:

1. Combine the first two terms:
\(-1 + 1 = 0\)

2. Now we have:
\[ 0 - 1 + 1 - 1 - 1 \]

3. Combine the next two terms:
\(-1 + 1 = 0\)

4. Now we have:
\[ 0 - 1 - 1 = -2 \]

### Final Result

Thus, the value of the expression is:

\[ \boxed{-2} \]