Complex Numbers
Product of Complex Numbers — Finding n
nta_pyq_2026_jan
Grade 11
Question:
Let $z=(1+i)(1+2i)(1+3i)\cdots(1+ni)$, where $i=\sqrt{-1}$. If $|z|^2=44200$, then $n$ is equal to _____
Step-by-Step Solution
Key Concept: $|z|^2=\prod_{k=1}^n|1+ki|^2=\prod_{k=1}^n(1+k^2)$. Compute the product: $(1+1)(1+4)(1+9)(1+16)(1+25)\cdots$
$\prod_{k=1}^5(1+k^2)=2\cdot5\cdot10\cdot17\cdot26=44200$. $n=5$.
Correct Answer: 5