The given relation `n^(n - 1) + n^(n - 1) = (n^2 +1)^(2) - (n^2 + 1)` is valid for every `n in N` if n equals to:

Options

A. a.3
B. b.5
C. c.1
D. d.both (a) and (c )

Answer

To solve the equation \( n^{(n - 1)} + n^{(n - 1)} = (n^2 + 1)^2 - (n^2 + 1) \) for \( n \in \mathbb{N} \), we can follow these steps:

### Step 1: Simplify the Left Side
The left side of the equation can be simplified:
\[ n^{(n - 1)} + n^{(n - 1)} = 2n^{(n - 1)} \]

### Step 2: Simplify the Right Side
Now, let's simplify the right side:
\[ (n^2 + 1)^2 - (n^2 + 1) \]
Expanding \( (n^2 + 1)^2 \):
\[ (n^2 + 1)(n^2 + 1) = n^4 + 2n^2 + 1 \]
Now, subtract \( (n^2 + 1) \):
\[ n^4 + 2n^2 + 1 - n^2 - 1 = n^4 + n^2 \]

### Step 3: Set the Two Sides Equal
Now we have:
\[ 2n^{(n - 1)} = n^4 + n^2 \]

### Step 4: Rearrange the Equation
Rearranging gives:
\[ 2n^{(n - 1)} - n^4 - n^2 = 0 \]

### Step 5: Test Natural Numbers
Now we will test natural numbers to find the values of \( n \) that satisfy this equation.

1. **For \( n = 1 \)**:
\[ 2(1^{(1 - 1)}) = 2(1^0) = 2 \]
\[ 1^4 + 1^2 = 1 + 1 = 2 \]
Both sides are equal.

2. **For \( n = 2 \)**:
\[ 2(2^{(2 - 1)}) = 2(2^1) = 4 \]
\[ 2^4 + 2^2 = 16 + 4 = 20 \]
Not equal.

3. **For \( n = 3 \)**:
\[ 2(3^{(3 - 1)}) = 2(3^2) = 2 \times 9 = 18 \]
\[ 3^4 + 3^2 = 81 + 9 = 90 \]
Not equal.

4. **For \( n = 4 \)**:
\[ 2(4^{(4 - 1)}) = 2(4^3) = 2 \times 64 = 128 \]
\[ 4^4 + 4^2 = 256 + 16 = 272 \]
Not equal.

5. **For \( n = 5 \)**:
\[ 2(5^{(5 - 1)}) = 2(5^4) = 2 \times 625 = 1250 \]
\[ 5^4 + 5^2 = 625 + 25 = 650 \]
Not equal.

### Conclusion
The only value of \( n \) that satisfies the equation is \( n = 1 \).

### Final Answer
The relation is valid for \( n = 1 \).