If z = (1 + i)(1 + 2i)(1 + 3i) .... (1 + ni) where n ∈ N and |z|² = 44200, then n is equal to

To find the value of $n$, we start by stating the given equation for the complex number $z$:

$z=(1+i)(1+2i)(1+3i)���(1+ni)$

We are given that the square of the modulus of $z$ is:

${\left|z\right|}^2=44200$

Recall the fundamental property of the modulus of complex numbers: the modulus of a product of complex numbers is equal to the product of their individual moduli. That is, for any complex numbers $z_1,z_2,\cdots ,z_n$:

$|z_1{z}_2\cdots z_n|=\left|z_1\right|\cdot \left|z_2\right|\cdots \left|z_n\right|$

Squaring both sides of this property gives:

${|z_1{z}_2\cdots z_n|}^2={\left|z_1\right|}^2\cdot {\left|z_2\right|}^2\cdots {\left|z_n\right|}^2$

Applying this to our expression for $z$:

${\left|z\right|}^2={|1+i|}^2\cdot {|1+2i|}^2\cdot {|1+3i|}^2\cdots {|1+ni|}^2$

For any complex number of the form $w=a+bi$, the square of its modulus is given by:

${\left|w\right|}^2=a^2+b^2$

We calculate the square of the modulus for each term in the product:

${|1+i|}^2=1^2+1^2=2$
${|1+2i|}^2=1^2+2^2=5$
${|1+3i|}^2=1^2+3^2=10$
$⋮$
${|1+ni|}^2=1+n^2$

Substituting these values back into the equation for ${\left|z\right|}^2$:

$2\cdot 5\cdot 10\cdots (1+n^2)=44200$

Let's evaluate the product step-by-step for successive values of $n$ to find when the product equals $44200$:

For $n=1$:
$\text{Product}=2$

For $n=2$:
$\text{Product}=2\cdot 5=10$

For $n=3$:
$\text{Product}=10\cdot (1+3^2)=10\cdot 10=100$

For $n=4$:
$\text{Product}=100\cdot (1+4^2)=100\cdot 17=1700$

For $n=5$:
$\text{Product}=1700\cdot (1+5^2)=1700\cdot 26=44200$

Thus, the relation is satisfied when $n=5$.