Q. Let
\[
z = (1+i)(1+2i)(1+3i)\ldots(1+ni),
\]
where \(i=\sqrt{-1}\).
If \(|z|^2 = 44200\), then \(n\) is equal to
Explanation
Taking modulus on both sides,
\[
|z| = |1+i||1+2i||1+3i|\ldots|1+ni|
\]
Using \(|a+bi| = \sqrt{a^2+b^2}\),
\[
|z|^2 = (1^2+1^2)(1^2+2^2)(1^2+3^2)\ldots(1^2+n^2)
\]
\[
|z|^2 = (2)(5)(10)\ldots(1+n^2)
\]
Checking values,
\[
2 \times 5 \times 10 \times 17 \times 26 = 44200
\]
This corresponds to
\[
n = 5
\]
Therefore, the correct value of \(n\) is 5.