donjar

Barisan floor-flooran

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Definisikan sebuah barisan $\{ a_n \}$ dimana \[ a_n = \frac{1}{n} \left( \left\lfloor \frac{n}{1} \right\rfloor + \left\lfloor \frac{n}{2} \right\rfloor + \dotsb + \left\lfloor \frac{n}{n} \right\rfloor \right). \] Buktikan terdapat tak berhingga banyaknya $n$ sehingga $a_{n + 1} > a_n$, dan juga terdapat tak berhingga banyaknya $n$ sehingga $a_{n + 1} < a_n$.

 

Sumber:

IMO Shortlist 2006 N3

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