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: Understanding the Concept:
For a complex number \(z = p + iq\), the product \(z\bar z = |z|^2 = p^2 + q^2\). Also, \(|z_1 z_2| = |z_1||z_2|\).

Step 2: Take the modulus squared:
Given \((1+i)(1+2i)\cdots(1+ni) = x + iy\).
Taking the squared modulus on both sides:
\[ |1+i|^2 |1+2i|^2 \cdots |1+ni|^2 = x^2 + y^2 \]

Step 3: Evaluate each factor:
\(|1+ki|^2 = 1 + k^2\). So the left side is
\[ (1+1)(1+4)(1+9)\cdots(1+n^2) = 2\cdot5\cdot10\cdots(1+n^2) \]
This is exactly the product asked for.

Step 4: Result:
The required product equals \(x^2 + y^2\). Option (D), \(x^2 - y^2\), would be the real part of the square, not the modulus squared. Options (A) and (C) involve square roots of \(x\) and \(y\), which do not arise.

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