If ` n gt 0 ` is an odd integer and ` x = (sqrt(2) + 1)^(n), f = x - [x], " then" (1 - f^(2))/(f) ` is

Options

A. an irrational number
B. a non-integer rational number
C. an odd number
D. an even number

Answer

To solve the problem step by step, we will follow the approach outlined in the video transcript.

### Step 1: Define the variables
Let \( n \) be an odd integer greater than 0. We define:
\[ x = (\sqrt{2} + 1)^n \]
and
\[ f = x - [x] \]
where \([x]\) is the nearest integer function.

**Hint:** Start by substituting a small odd integer for \( n \) to simplify calculations.

### Step 2: Choose a value for \( n \)
Let’s choose \( n = 1 \) (since it is an odd integer):
\[ x = \sqrt{2} + 1 \]

**Hint:** Choosing \( n = 1 \) makes calculations easier and helps to understand the pattern.

### Step 3: Calculate \( f \)
Now, we compute \( f \):
\[ f = x - [x] = (\sqrt{2} + 1) - [\sqrt{2} + 1] \]
Calculating \(\sqrt{2}\):
\[ \sqrt{2} \approx 1.41 \quad \Rightarrow \quad \sqrt{2} + 1 \approx 2.41 \]
Thus, the nearest integer \([x]\) is 2. Therefore:
\[ f = (\sqrt{2} + 1) - 2 = \sqrt{2} - 1 \]

**Hint:** Remember that \([x]\) is the greatest integer less than or equal to \( x \).

### Step 4: Calculate \( f^2 \)
Next, we need to compute \( f^2 \):
\[ f^2 = (\sqrt{2} - 1)^2 = 2 - 2\sqrt{2} + 1 = 3 - 2\sqrt{2} \]

**Hint:** Use the identity \((a - b)^2 = a^2 - 2ab + b^2\).

### Step 5: Substitute \( f \) and \( f^2 \) into the expression
Now, we need to evaluate:
\[ z = \frac{1 - f^2}{f} \]
Substituting the values we have:
\[ z = \frac{1 - (3 - 2\sqrt{2})}{\sqrt{2} - 1} \]
This simplifies to:
\[ z = \frac{1 - 3 + 2\sqrt{2}}{\sqrt{2} - 1} = \frac{-2 + 2\sqrt{2}}{\sqrt{2} - 1} \]

**Hint:** Combine like terms carefully when substituting.

### Step 6: Simplify the expression
Factor out a 2 from the numerator:
\[ z = \frac{2(\sqrt{2} - 1)}{\sqrt{2} - 1} \]
Since \(\sqrt{2} - 1 \neq 0\), we can cancel:
\[ z = 2 \]

**Hint:** Ensure you check that the denominator is not zero before canceling.

### Step 7: Determine the nature of \( z \)
The final value of \( z \) is 2, which is an even number.

**Hint:** Review the properties of numbers to classify the result.

### Conclusion
Thus, the expression \( \frac{1 - f^2}{f} \) evaluates to an even number.

**Final Answer:** The expression \( \frac{1 - f^2}{f} \) is an even number.