Question

If (1+i)(1+2i)(1+3i).....(1+ni)=x+iy, then 2.5.10.....(1+n2) = ?

Answer 1

Answer:

$\bf\,2.5.10.....(1+n^2)=x^2+y^2$

Step-by-step explanation:

Given:

$(1+i)(1+2i)(1+3i).....(1+ni)=x+iy$

Taking modulus on both sides, we get

$|(1+i)(1+2i)(1+3i).....(1+ni)|=|x+iy|$

Using the property,

$\boxed{|z_1\,z_2\,z_3......z_n|=|z_1|\,|z_2|\,|z_3|.......|z_n|}$

$|1+i|\:|1+2i|\:|1+3i|.....|1+ni|=|x+iy|$

$\sqrt{1^2+1^2}\:\sqrt{1^2+2^2}\:\sqrt{1^2+3^2}.....\sqrt{1^2+n^2}=\sqrt{x^2+y^2}$

$\sqrt{1+1}\:\sqrt{1+4}\:\sqrt{1+9}.....\sqrt{1+n^2}=\sqrt{x^2+y^2}$

$\sqrt{2}\:\sqrt{5}\:\sqrt{10}.....\sqrt{1+n^2}=\sqrt{x^2+y^2}$

squarring on both sides

$\implies\boxed{\bf\,2.5.10.....(1+n^2)=x^2+y^2}$