Răspuns:
[tex]\boldsymbol{ \red{x = 2019^2}}[/tex]
Explicație pas cu pas:
La numărător utilizăm formula:
[tex]\boxed{\boldsymbol{\dfrac{1}{n \cdot (n + 1)} = \dfrac{1}{n} - \dfrac{1}{n + 1}}}[/tex]
[tex]2 \bigg(\dfrac{1}{1 \cdot2} + \dfrac{1}{2\cdot3} + \dfrac{1}{3\cdot4} + ... + \dfrac{1}{2019 \cdot 2020}\bigg) =\\[/tex]
[tex]= 2 \bigg(\dfrac{1}{1} - \dfrac{1}{2} + \dfrac{1}{2} - \dfrac{1}{3} + \dfrac{1}{3} - \dfrac{1}{4} + ... + \dfrac{1}{2018} - \dfrac{1}{2019} + \dfrac{1}{2019} - \dfrac{1}{2020} \bigg) \\[/tex]
[tex]= 2 \bigg(\dfrac{1}{1} - \dfrac{1}{2020} \bigg) = 2 \cdot \dfrac{2020 - 1}{2020} = \dfrac{2019}{1010}\\[/tex]
La numitor avem o sumă Gauss:
[tex]\boxed {\boldsymbol{1 + 2 + 3 + ... + n = \dfrac{n \cdot (n + 1)}{2}}}[/tex]
[tex]\dfrac{1}{1 + 2 + 3 + 4 + ... + 2018 + 2019} = \\[/tex]
[tex]= \dfrac{1}{\dfrac{2019 \cdot (2019 + 1)}{2} } = \dfrac{1}{\dfrac{2019 \cdot 2020}{2} } = \dfrac{1}{2019 \cdot 1010}\\[/tex]
Numărul x este:
[tex]x = \dfrac{\dfrac{2019}{1010}}{\dfrac{1}{2019 \cdot 1010}} = \dfrac{2019}{1010} : \dfrac{1}{2019 \cdot 1010} = \\[/tex]
[tex]= \dfrac{2019}{1010} \cdot \dfrac{2019 \cdot 1010}{1} = \bf2019^2[/tex]
⇒ x este pătratul unui număr natural
