If $(1+\mathrm{i})(1+2 \mathrm{i})(1+3 \mathrm{i}) \ldots .(1+\mathrm{ni})=\mathrm{x}+$ iy then $2.5

Question

If $(1+\mathrm{i})(1+2 \mathrm{i})(1+3 \mathrm{i}) \ldots .(1+\mathrm{ni})=\mathrm{x}+$ iy then $2.5 .10 \ldots .\left(1+n^2\right)=$ $\_\_\_\_$

Professor PC Thomas Classes · Answer

Taking the modulus on both sides and squaring gives $(1^2+1^2)(1^2+2^2)\dots(1^2+n^2) = x^2 + y^2$. So it equals $x^2 + y^2$.