If the number of terms in ` (x + 1 + (1)/(x))^(n) (n in I ^(+) " is " 401 `,
then n is greater then

Options

A. 201
B. 200
C. 199
D. None of these

Explanation

To solve the problem, we need to find the value of \( n \) such that the number of terms in the expression \( (x + 1 + \frac{1}{x})^n \) is 401. Let's go through the steps systematically.

### Step 1: Understand the expression
The expression we have is \( (x + 1 + \frac{1}{x})^n \). To find the number of terms in this expansion, we can rewrite it in a more manageable form.

### Step 2: Rewrite the expression
We can express \( (x + 1 + \frac{1}{x})^n \) as follows:
\[ (x + 1 + \frac{1}{x})^n = \left(1 + x + x^2\right)^n \cdot \frac{1}{x^n} \]
This shows that we can focus on the polynomial \( (1 + x + x^2)^n \).

### Step 3: Determine the number of terms in the polynomial
The polynomial \( (1 + x + x^2)^n \) can be analyzed for the number of distinct terms it produces. The number of distinct terms in the expansion of \( (1 + x + x^2)^n \) can be determined by the number of different powers of \( x \) that can be formed.

### Step 4: Find the range of powers
The minimum power of \( x \) occurs when we take \( n \) terms of \( 1 \) (which gives \( 0 \)), and the maximum power occurs when we take \( n \) terms of \( x^2 \) (which gives \( 2n \)). Therefore, the powers of \( x \) in the expansion range from \( 0 \) to \( 2n \).

### Step 5: Calculate the number of distinct terms
The number of distinct terms in the polynomial \( (1 + x + x^2)^n \) is given by the number of integer values from \( 0 \) to \( 2n \), which is:
\[ 2n + 1 \]
This is because the powers of \( x \) can take on every integer value from \( 0 \) to \( 2n \).

### Step 6: Set up the equation
According to the problem, the number of terms is given as 401. Therefore, we can set up the equation:
\[ 2n + 1 = 401 \]

### Step 7: Solve for \( n \)
Now, we solve for \( n \):
\[ 2n = 401 - 1 \]
\[ 2n = 400 \]
\[ n = \frac{400}{2} = 200 \]

### Step 8: Determine the condition for \( n \)
The problem asks for \( n \) to be greater than a certain value. Since we found \( n = 200 \), we conclude:
\[ n > 199 \]

Thus, the final answer is:
\[ \text{n is greater than } 199. \]

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