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Omni-MATH / Suppose a₁,a₂,…,a₁₀₀ are positive real numbers such that a_(k)=(ka_(k−1))/(a_(k−1)−(k−1)) for k=2,3,…,100. Given that a₂₀=a₂₃, compute a₁₀₀.

Problem

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problem

Suppose a1,a2,…,a100a_{1}, a_{2}, \ldots, a_{100} are positive real numbers such that ak=kak−1ak−1−(k−1)a_{k}=\frac{k a_{k-1}}{a_{k-1}-(k-1)} for k=2,3,…,100k=2,3, \ldots, 100. Given that a20=a23a_{20}=a_{23}, compute a100a_{100}.
Plain-text mathematical notation (without MathML)
Suppose a₁,a₂,…,a₁₀₀ are positive real numbers such that a_(k)=(ka_(k−1))/(a_(k−1)−(k−1)) for k=2,3,…,100. Given that a₂₀=a₂₃, compute a₁₀₀.
Original LaTeX notation
Suppose $a_{1}, a_{2}, \ldots, a_{100}$ are positive real numbers such that $$a_{k}=\frac{k a_{k-1}}{a_{k-1}-(k-1)}$$ for $k=2,3, \ldots, 100$. Given that $a_{20}=a_{23}$, compute $a_{100}$.

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Source and history

task_id
15b08fd0-761f-590f-95df-acdb76b22508
task_key
test--15b08fd0-761f-590f-95df-acdb76b22508
task_revision_id
1
config / split
/ test

Official source

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