Choose the statement which is correct for all `n in N`

Options

A. `2n gt 2n+1`
B. `x^(n)-y^(n)` is divisible by (x+y)
C. `x^(n)-y^(n)` is divisible by (x-y)
D. `1+(1)/(4)+(1)/(9)+(1)/(16)+....+(1)/(n^(2)) lt 2-(1)/(n)`

Answer

To solve the question, we need to analyze the given statements one by one and determine which one is correct for all natural numbers \( n \in \mathbb{N} \).

### Step 1: Analyze the First Statement
**Statement 1:** \( 2n > 2n + 1 \)

- **Substituting \( n = 1 \):**
- Left-hand side (LHS): \( 2 \times 1 = 2 \)
- Right-hand side (RHS): \( 2 \times 1 + 1 = 3 \)

We check if \( 2 > 3 \), which is false. Thus, this statement is incorrect for \( n = 1 \) and therefore false for all natural numbers.

### Step 2: Analyze the Second Statement
**Statement 2:** \( x^n - y^n \) is divisible by \( x + y \)

- **Substituting \( n = 1 \):**
- The statement becomes \( x^1 - y^1 = x - y \).

We need to check if \( x - y \) is divisible by \( x + y \). This is not necessarily true for all \( x \) and \( y \) (e.g., if \( x = 1 \) and \( y = 1 \), \( 0 \) is divisible by \( 2 \), but if \( x = 2 \) and \( y = 1 \), \( 1 \) is not divisible by \( 3 \)). Thus, this statement is false.

### Step 3: Analyze the Third Statement
**Statement 3:** \( x^n - y^n \) is divisible by \( x - y \)

- **Substituting \( n = 1 \):**
- The statement becomes \( x^1 - y^1 = x - y \).

Clearly, \( x - y \) is divisible by \( x - y \) (it is equal). This is true.

- **Substituting \( n = 2 \):**
- The statement becomes \( x^2 - y^2 = (x - y)(x + y) \).

Since \( (x - y)(x + y) \) is divisible by \( x - y \), this is true.

- **Substituting \( n = 3 \):**
- The statement becomes \( x^3 - y^3 = (x - y)(x^2 + xy + y^2) \).

Again, \( (x - y)(x^2 + xy + y^2) \) is divisible by \( x - y \), so this is true.

- **Inductive Step:**
Assuming it holds for \( n = k \), we can show it holds for \( n = k + 1 \) using the factorization of \( x^{k+1} - y^{k+1} \). Thus, by induction, this statement is true for all natural numbers \( n \).

### Step 4: Analyze the Fourth Statement
**Statement 4:** \( 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \ldots + \frac{1}{n^2} < 2 - \frac{1}{n} \)

- **Substituting \( n = 1 \):**
- LHS: \( 1 \)
- RHS: \( 2 - 1 = 1 \)

We check if \( 1 < 1 \), which is false.

Thus, this statement is also false.

### Conclusion
The only statement that holds true for all natural numbers \( n \) is the **third statement**: \( x^n - y^n \) is divisible by \( x - y \).