If ` ( 1+i) (1+ 2i) (1+ 3i) …(1 +ni) = x+ iy, `then ` 2,5,10 ….( 1+n^(2)) = x^(k) +y^(k) ` The value of k is :

Options

A. 1
B. 2
C. 4
D. None of these

Answer

To solve the given problem step by step, we will analyze the expression and derive the value of \( k \).

### Step 1: Understand the Given Expression
We start with the product:
\[ (1+i)(1+2i)(1+3i)\ldots(1+ni) = x + iy \]
This means that the product of these complex numbers results in a complex number represented as \( x + iy \).

**Hint:** Recognize that \( x \) and \( y \) are the real and imaginary parts of the product, respectively.

### Step 2: Take the Conjugate
Next, we take the conjugate of both sides:
\[ (1-i)(1-2i)(1-3i)\ldots(1-ni) = x - iy \]
This gives us a new equation that we will refer to as Equation 2.

**Hint:** The conjugate of a complex number \( a + bi \) is \( a - bi \).

### Step 3: Multiply the Two Equations
Now, we multiply Equation 1 and Equation 2:
\[ [(1+i)(1-1i)][(1+2i)(1-2i)]\ldots[(1+ni)(1-ni)] = (x + iy)(x - iy) \]
The left-hand side simplifies to:
\[ (1^2 - i^2)(1^2 - (2i)^2)(1^2 - (3i)^2)\ldots(1^2 - (ni)^2) \]
And the right-hand side simplifies to:
\[ x^2 + y^2 \]

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

### Step 4: Simplify Each Factor
Each factor on the left-hand side becomes:
\[ 1^2 - (ki)^2 = 1 + k^2 \quad \text{for } k = 1, 2, \ldots, n \]
Thus, we have:
\[ (1 + 1^2)(1 + 2^2)(1 + 3^2)\ldots(1 + n^2) \]

**Hint:** Recognize that \( i^2 = -1 \) and thus \( -(-1) = +1 \).

### Step 5: Calculate the Product
The product becomes:
\[ 2 \cdot (1 + 1^2)(1 + 2^2)(1 + 3^2)\ldots(1 + n^2) = x^2 + y^2 \]
The left-hand side can be expressed as:
\[ 2 \cdot (1 + 1)(1 + 4)(1 + 9)\ldots(1 + n^2) = 2 \cdot (2)(5)(10)\ldots(1 + n^2) \]

**Hint:** The series \( 2, 5, 10, \ldots \) can be derived from the factors \( 1 + k^2 \).

### Step 6: Relate to \( x^k + y^k \)
We know from the problem statement that:
\[ 2, 5, 10, \ldots, (1 + n^2) = x^k + y^k \]
From our earlier simplification, we have:
\[ x^2 + y^2 \]
This implies that \( k = 2 \).

**Hint:** Compare the forms of the two expressions to find the value of \( k \).

### Conclusion
Thus, the value of \( k \) is:
\[ \boxed{2} \]