Understanding Complex Number Magnitude

We are given a complex number $z$ defined as a product of terms: $z = (1 + i)(1 + 2i)(1 + 3i) \dots (1 + ni)$ where $i = \sqrt{-1}$. We are also given that the square of its magnitude is $|z|^2 = 44200$. The objective is to find the value of $n$.

Modulus Property for Products

A key property of complex number magnitudes is that the magnitude of a product is the product of the magnitudes.

Therefore, $|z| = |1 + i| \cdot |1 + 2i| \cdot |1 + 3i| \dots |1 + ni|$.

The magnitude of a complex number $a + bi$ is calculated as $|a + bi| = \sqrt{a^2 + b^2}$.

Applying this, we get:

$|z| = \sqrt{1^2 + 1^2} \cdot \sqrt{1^2 + 2^2} \cdot \sqrt{1^2 + 3^2} \dots \sqrt{1^2 + n^2}$

This simplifies to:

$|z| = \sqrt{(1+1^2)(1+2^2)(1+3^2) \dots (1+n^2)}$

Calculating |z|^2

Squaring both sides gives the magnitude squared:

$|z|^2 = (1+1^2)(1+2^2)(1+3^2) \dots (1+n^2)$

We are given $|z|^2 = 44200$. So, we need to find $n$ such that:

$ \prod_{k=1}^{n} (1+k^2) = 44200 $

Step-by-Step Calculation

Let's compute the product for successive values of $n$:

• For $n=1$: $|z|^2 = (1+1^2) = 2$
• For $n=2$: $|z|^2 = (1+1^2)(1+2^2) = 2 \times (1+4) = 2 \times 5 = 10$
• For $n=3$: $|z|^2 = (1+1^2)(1+2^2)(1+3^2) = 10 \times (1+9) = 10 \times 10 = 100$
• For $n=4$: $|z|^2 = (1+1^2)(1+2^2)(1+3^2)(1+4^2) = 100 \times (1+16) = 100 \times 17 = 1700$
• For $n=5$: $|z|^2 = (1+1^2)(1+2^2)(1+3^2)(1+4^2)(1+5^2) = 1700 \times (1+25) = 1700 \times 26 = 44200$

The calculated value for $n=5$ matches the given $|z|^2 = 44200$.

Conclusion

The value of $n$ that satisfies the condition $|z|^2 = 44200$ is $n=5$.