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OlympiadBench / 2032 / Let a₀,a₁,a₂,… be a sequence of real numbers such that…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

question

Let

a_{0}, a_{1}, a_{2}, \ldots

be a sequence of real numbers such that

a_{0}=0, a_{1}=1

, and for every

n \geqslant 2

there exists

1 \leqslant k \leqslant n

satisfying

a_{n}=\frac{a_{n-1}+\cdots+a_{n-k}}{k}

Find the maximal possible value of

a_{2018}-a_{2017}

Plain-text mathematical notation (without MathML)

Let a₀,a₁,a₂,… be a sequence of real numbers such that a₀=0,a₁=1, and for every n≥2 there exists 1≤k≤n satisfying

a_(n)=(a_(n−1)+⋯+a_(n−k))/(k)

Find the maximal possible value of a₂₀₁₈−a₂₀₁₇.

Original LaTeX notation

Let $a_{0}, a_{1}, a_{2}, \ldots$ be a sequence of real numbers such that $a_{0}=0, a_{1}=1$, and for every $n \geqslant 2$ there exists $1 \leqslant k \leqslant n$ satisfying

$$
a_{n}=\frac{a_{n-1}+\cdots+a_{n-k}}{k}
$$

Find the maximal possible value of $a_{2018}-a_{2017}$.

answer type

Numerical

is multiple answer

false

language

English

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID: 2032
task_id: f7b86486-f5dd-5aae-9092-1c9803138d9e
task_key: OE~5fTO~5fmaths~5fen~5fCOMP--train--2032
task_revision_id: 3
config / split: OE_TO_maths_en_COMP / train

Official source

initial import