Let z = (1 + i)(1 + 2i)(1 + 3i) . . . (1 + ni), where i = √−1. If |z|² = 44200, then n is equal to _____.

Let z = (1 + i)(1 + 2i)(1 + 3i) … (1 + ni) where i = √−1. If |z|² = 44200, then n is equal to

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

Correct Answer: 5

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.

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