Question

If \((1+i)\cdot (1+2i)\ldots \ldots \ldots (1+ni) = x+iy\) (Where \(i = \sqrt{-1}\) ), then the value of \((2)\cdot (5)\cdot (10)\ldots \ldots \ldots (1+n^2)\)

• A. \(\frac{\sqrt{x}+\sqrt{y}}{2}\)
• B. \(x^2+y^2\)
• C. \(\frac{\sqrt{x}+\sqrt{y}}{\sqrt{x}-\sqrt{y}}\)
• D. \(x^2-y^2\)

Hint:

Take the modulus squared on both sides of the given product.

Answer 1

The Correct Option is B

Step 1: Conjugate method:
Multiply the given product by its conjugate product: $(x+iy)(x-iy) = x^2 + y^2$.

Step 2: Pair the factors:
Each factor $(1+ki)$ pairs with $(1-ki)$ to give $(1+ki)(1-ki) = 1 + k^2$.

Step 3: Conclude:
So $x^2 + y^2 = \prod_{k=1}^{n}(1+k^2) = 2\cdot5\cdot10\cdots(1+n^2)$, option (B).

Final Answer:
The product equals x^2 + y^2. \[ \boxed{\text{(B) }x^2+y^2} \]