Question

If (2 + i) (2 + 2i) (2 + 3i) ...... (2 + ni) = x + iy, then 5.8.13...(4 + n²) = ____________.

Answer 1

Given:- (2 + i) (2 + 2i) (2 + 3i) ...... (2 + ni) = x + iy
Taking modulus both sides, we get
|(2 + i) (2 + 2i) (2 + 3i) ...... (2 + ni) = |x + iy|
By squaring both sides, we get
$$\begin{gathered} {\left|2+i\right|}^2{\left|2+2i\right|}^2{\left|2+3i\right|}^2......{\left|2+ni\right|}^2={\left|x+iy\right|}^2 \\ \mathrm{i}.\mathrm{e}\left(4+1\right)\left(4+4\right)\left(4+9\right)......\left(4+n^2\right)=x^2+y^2 \\ \mathrm{i}.\mathrm{e}5.8.13......\left(4+n^2\right)=x^2+y^2 \end{gathered}$$