Let z C with Im(z)=10, satisfying 2z-n2z+n=2i-1 for some natural number n.

Question

Let $z\in\mathbb C$ with $\operatorname{Im}(z)=10$, satisfying $\dfrac{2z-n}{2z+n}=2i-1$ for some natural number $n$. Then:

Accepted Answer

Let $\operatorname{Re}(z)=x$, so $z=x+10i$. $2z-n=(2i-1)(2z+n)$: $(2x-n)+20i=(2i-1)[(2x+n)+20i]$.

Comparing real/imaginary parts: $4x=-40 \implies x=-10$, and $40=-40+2n \implies n=40$.

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