If $(1 + i) (1 + 2 i) (1 + 3 i) . . . . (1 + n i) = x + i y$ , show that $2.5.10..... (1 + n^{2}) = x^{2} + y^{2}$

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Solution

We have,
$\begin{matrix} (1 + i) (1 + 2 i) (1 + 3 i) . . . . . . . . . . (1 + n i) = x + i y \Rightarrow | (1 + i) (1 + 2 i) (1 + 3 i) . . . . . . . (1 + n i) | = \sqrt{x^{2} + y^{2}} \Rightarrow∴ \sqrt{2} . \sqrt{5} . \sqrt{10} . . . . . . . . . . . . \sqrt{1 + n^{2}} = \sqrt{x^{2} + y^{2}} O n s q u a r i n g b o t h s i d e \Rightarrow∴ 2.5.10........ (1 + n^{2}) = x^{2} + y^{2} \end{matrix}$

Hence, proved.

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