Explicație pas cu pas:
(cu o modificare de enunț, la final este 100 în loc de 99)
[tex]x = \Big(\frac{1}{1} + \frac{1}{1 + 2} + \frac{1}{1 + 2 + 3} + ... + \frac{1}{1 + 2 + 3 + ... + 100} \Big) \cdot \frac{101}{100} \\ [/tex]
[tex]= \Big(\frac{1}{1} + \frac{1}{ \frac{2 \cdot 3}{2} } + \frac{1}{ \frac{3 \cdot 4}{2} } + ... + \frac{1}{ \frac{100 \cdot 101}{2} } \Big) \cdot \frac{101}{100} \\ [/tex]
[tex]= \Big(\frac{2}{2} + \frac{2}{2 \cdot 3} + \frac{2}{3 \cdot 4} + ... + \frac{2}{100 \cdot 101} \Big) \cdot \frac{101}{100} \\ [/tex]
[tex]= 2 \cdot \Big( \frac{1}{1} - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + ... + \frac{1}{100} - \frac{1}{101} \Big) \cdot \frac{101}{100} \\ [/tex]
[tex]= 2 \cdot \Big( \frac{1}{1} - \frac{1}{101} \Big) \cdot \frac{101}{100} \\ [/tex]
[tex]= 2 \cdot \frac{101 - 1}{101} \cdot \frac{101}{100} \\ [/tex]
[tex]= 2 \cdot \frac{100}{101} \cdot \frac{101}{100} = \bf 2 \in \mathbb{N}\\ [/tex]
