Solve the following
Question:

If $(1+i)(1+2 i)(1+3 i) \ldots(1+n i)=a+i b$, then $2 \times 5 \times 10 \times \ldots \times\left(1+n^{2}\right)$ is equal to

(a) $\sqrt{a^{2}+b^{2}}$

(b) $\sqrt{a^{2}-b^{2}}$

(c) $a^{2}+b^{2}$

(d) $a^{2}-b^{2}$

(e) $a+b$

Solution:

(c) $a^{2}+b^{2}$

$(1+i)(1+2 i)(1+3 i) \ldots \ldots(1+n i)=a+i b$

Taking modulus on both the sides, we get:

$|(1+i)(1+2 i)(1+3 i) \ldots \ldots(1+n i)|=|a+i b|$

$|(1+i)(1+2 i)(1+3 i) \ldots \ldots(1+n i)|$ can be written as $|(1+i)||(1+2 i)||(1+3 i)| \ldots \ldots \ldots|(1+n i)|$

$\sqrt{1^{2}+1^{2}} \times \sqrt{1^{2}+2^{2}} \times \sqrt{1^{2}+3^{2}} \times \ldots \times \sqrt{1+n^{2}}=\sqrt{a^{2}+b^{2}}$

$\Rightarrow \sqrt{2} \times \sqrt{5} \times \sqrt{10} \times \ldots \times \sqrt{1+n^{2}}=\sqrt{a^{2}+b^{2}}$

Squaring on both the sides, we get:

$2 \times 5 \times 10 \times \ldots \times\left(1+n^{2}\right)=a^{2}+b^{2}$

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