The period of `f(x)=sin((pix)/(n-1))+ cos ((pix)/(n)), n in Z, n gt 2`, is

Options

A. `2 n pi (n-1)`
B. `4 (n-1)pi `
C. `2n(n-1)`
D. none of these

Answer

To find the period of the function \( f(x) = \sin\left(\frac{\pi x}{n-1}\right) + \cos\left(\frac{\pi x}{n}\right) \), where \( n \in \mathbb{Z} \) and \( n > 2 \), we can follow these steps:

### Step 1: Identify the periods of the individual components
The function \( f(x) \) is composed of two periodic functions: \( \sin\left(\frac{\pi x}{n-1}\right) \) and \( \cos\left(\frac{\pi x}{n}\right) \).

- The period of \( \sin(kx) \) is given by \( \frac{2\pi}{k} \).
- For \( \sin\left(\frac{\pi x}{n-1}\right) \), we have:
\[ k = \frac{\pi}{n-1} \implies \text{Period } T_1 = \frac{2\pi}{\frac{\pi}{n-1}} = 2(n-1) \]

- The period of \( \cos(kx) \) is also given by \( \frac{2\pi}{k} \).
- For \( \cos\left(\frac{\pi x}{n}\right) \), we have:
\[ k = \frac{\pi}{n} \implies \text{Period } T_2 = \frac{2\pi}{\frac{\pi}{n}} = 2n \]

### Step 2: Find the least common multiple (LCM) of the periods
The overall period of the function \( f(x) \) will be the least common multiple of the two individual periods \( T_1 \) and \( T_2 \):
\[ \text{Period of } f(x) = \text{LCM}(T_1, T_2) = \text{LCM}(2(n-1), 2n) \]

### Step 3: Simplify the LCM
Since both periods have a common factor of 2, we can factor that out:
\[ \text{LCM}(2(n-1), 2n) = 2 \cdot \text{LCM}(n-1, n) \]

### Step 4: Determine LCM of \( n-1 \) and \( n \)
The numbers \( n-1 \) and \( n \) are consecutive integers. The LCM of two consecutive integers is always their product:
\[ \text{LCM}(n-1, n) = (n-1) \cdot n \]

### Step 5: Combine the results
Thus, we have:
\[ \text{Period of } f(x) = 2 \cdot (n-1) \cdot n = 2n(n-1) \]

### Final Answer
The period of the function \( f(x) \) is:
\[ \boxed{2n(n-1)} \]