Question

(50 PONTOS) Considere a seguinte sequência numérica:
$~$
$B_{n}=\frac{1}{2}\,\mathrm{cossec}(\frac{1}{2})\cdot n\,\mathrm{sen}(n-\frac{1}{2})+\frac{1}{4}\,\mathrm{cossec^{2}}(\frac{1}{2})\cdot \cos n$

com $n$ natural.

a) Mostre que $\Delta(B_{n})=B_{n+1}-B_{n}=n\cos n.$

b) Calcule $\displaystyle\sum\limits_{n=0}^{100}{n\cos n}.$

Answer 1

Considere as sequências

$a_{n}=n\cdot sen(n-\frac{1}{2})\\\\b_{n}=cos(n)$

E veja as resoluções em anexo
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b)

Encontrando x₁₀₁:

$x_{101}=\frac{1}{2}csc(\frac{1}{2})\cdot101\cdot sen(101-\frac{1}{2})+\frac{1}{4}csc^{2}(\frac{1}{2})\cdot cos(101)\\\\\boxed{\boxed{x_{101}=\frac{101}{2}csc\left(\frac{1}{2}\right)sen\left(\frac{201}{2}\right)+\frac{1}{4}csc^{2}\left(\frac{1}{2}\right)cos(101)}}$

Encontrando x₀:

$x_{0}=\frac{1}{2}csc(\frac{1}{2})\cdot0\cdot sen(0-\frac{1}{2})+\frac{1}{4}csc^{2}(\frac{1}{2})cos(0)\\\\x_{0}=0+\frac{1}{4}csc^{2}(\frac{1}{2})\cdot1\\\\\boxed{\boxed{x_{0}=\frac{1}{4}csc^{2}\left(\frac{1}{2}\right)}}$

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