If (2 + i) (2 + 2i) (2 + 3i) ...... (2 + ni) = x + iy,

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

$|2+i|^{2}|2+2 i|^{2}|2+3 i|^{2} \ldots \ldots|2+n i|^{2}=|x+i y|^{2}$

i.e $(4+1)(4+4)(4+9) \ldots \ldots\left(4+n^{2}\right)=x^{2}+y^{2}$

i.e $5.8 .13 \ldots \ldots\left(4+n^{2}\right)=x^{2}+y^{2}$