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Omni-MATH / A sequence (a_(n)) of real numbers is defined by a₀=1, a₁=2015 and for all n≥1, we have a_(n+1)=(n−1)/(n+1)a_(n)−(n−2)/(n²+n)a_(n−1). Calculate the value of …

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problem

A sequence (an)(a_n) of real numbers is defined by a0=1a_0=1, a1=2015a_1=2015 and for all n≥1n\geq1, we have an+1=n−1n+1an−n−2n2+nan−1.a_{n+1}=\frac{n-1}{n+1}a_n-\frac{n-2}{n^2+n}a_{n-1}. Calculate the value of a1a2−a2a3+a3a4−a4a5+…+a2013a2014−a2014a2015\frac{a_1}{a_2}-\frac{a_2}{a_3}+\frac{a_3}{a_4}-\frac{a_4}{a_5}+\ldots+\frac{a_{2013}}{a_{2014}}-\frac{a_{2014}}{a_{2015}}.
Plain-text mathematical notation (without MathML)
A sequence (a_(n)) of real numbers is defined by a₀=1, a₁=2015 and for all n≥1, we have
a_(n+1)=(n−1)/(n+1)a_(n)−(n−2)/(n²+n)a_(n−1).
Calculate the value of (a₁)/(a₂)−(a₂)/(a₃)+(a₃)/(a₄)−(a₄)/(a₅)+…+(a₂₀₁₃)/(a₂₀₁₄)−(a₂₀₁₄)/(a₂₀₁₅).
Original LaTeX notation
A sequence $(a_n)$ of real numbers is defined by $a_0=1$, $a_1=2015$ and for all $n\geq1$, we have
$$a_{n+1}=\frac{n-1}{n+1}a_n-\frac{n-2}{n^2+n}a_{n-1}.$$
Calculate the value of $\frac{a_1}{a_2}-\frac{a_2}{a_3}+\frac{a_3}{a_4}-\frac{a_4}{a_5}+\ldots+\frac{a_{2013}}{a_{2014}}-\frac{a_{2014}}{a_{2015}}$.

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Source and history

task_id
a105585d-dc90-5ac3-b3f6-292bb9620497
task_key
test--a105585d-dc90-5ac3-b3f6-292bb9620497
task_revision_id
1
config / split
/ test

Official source

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