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Omni-MATH / Let {z_(n)}_(n≥1) be a sequence of complex numbers, whose odd terms are real, even terms are purely imaginary, and for every positive integer k, |z_(k)z_(k+1)|=2^(k). Denote …

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problem

Let

\{ z_n \}_{n \ge 1}

be a sequence of complex numbers, whose odd terms are real, even terms are purely imaginary, and for every positive integer

k

|z_k z_{k+1}|=2^k

. Denote

f_n=|z_1+z_2+\cdots+z_n|,

for

n=1,2,\cdots

(1) Find the minimum of

f_{2020}

. (2) Find the minimum of

f_{2020} \cdot f_{2021}

Plain-text mathematical notation (without MathML)

Let {z_(n)}_(n≥1) be a sequence of complex numbers, whose odd terms are real, even terms are purely imaginary, and for every positive integer k, |z_(k)z_(k+1)|=2^(k). Denote f_(n)=|z₁+z₂+⋯+z_(n)|, for n=1,2,⋯
(1) Find the minimum of f₂₀₂₀.
(2) Find the minimum of f₂₀₂₀⋅f₂₀₂₁.

Original LaTeX notation

Let $\{ z_n \}_{n \ge 1}$ be a sequence of complex numbers, whose odd terms are real, even terms are purely imaginary, and for every positive integer $k$, $|z_k z_{k+1}|=2^k$. Denote $f_n=|z_1+z_2+\cdots+z_n|,$ for $n=1,2,\cdots$
(1) Find the minimum of $f_{2020}$.
(2) Find the minimum of $f_{2020} \cdot f_{2021}$.

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task_id: f4937414-671f-5112-8c43-19532a1aec5e
task_key: test--f4937414-671f-5112-8c43-19532a1aec5e
task_revision_id: 4
config / split: / test

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