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Omni-MATH / Define the sequence {x_(i)}_(i≥0) by x₀=2009 and x_(n)=−(2009)/(n)∑_(k=0)^(n−1)x_(k) for all n≥1. Compute the value of ∑_(n=0)²⁰⁰⁹2^(n)x_(n)

Problem

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problem

Define the sequence {xi}i≥0\{x_{i}\}_{i \geq 0} by x0=2009x_{0}=2009 and xn=−2009n∑k=0n−1xkx_{n}=-\frac{2009}{n} \sum_{k=0}^{n-1} x_{k} for all n≥1n \geq 1. Compute the value of ∑n=020092nxn\sum_{n=0}^{2009} 2^{n} x_{n}
Plain-text mathematical notation (without MathML)
Define the sequence {x_(i)}_(i≥0) by x₀=2009 and x_(n)=−(2009)/(n)∑_(k=0)^(n−1)x_(k) for all n≥1. Compute the value of ∑_(n=0)²⁰⁰⁹2^(n)x_(n)
Original LaTeX notation
Define the sequence $\{x_{i}\}_{i \geq 0}$ by $x_{0}=2009$ and $x_{n}=-\frac{2009}{n} \sum_{k=0}^{n-1} x_{k}$ for all $n \geq 1$. Compute the value of $\sum_{n=0}^{2009} 2^{n} x_{n}$

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Source and history

task_id
75eda9a2-706e-5cdc-996f-3b904114866a
task_key
test--75eda9a2-706e-5cdc-996f-3b904114866a
task_revision_id
1
config / split
/ test

Official source

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