Răspuns:
n=2020² -1
Explicație pas cu pas:
amplificand cu conjugat , vom obtine numitorii 1
rama
√2-1+√3-√2+.....+√n-√(n-1)+√(n+1)-√n= ...dupared term aemenea
√(n+1)-1=2019
√(n+1)=2020
n+1=2020²
n=2020² -1=..vezi tu, pe calculator
[tex]\it \dfrac{1}{\sqrt{n+1}+\sqrt n}=\dfrac{\sqrt{n+1}-\sqrt n}{n+1-n}=\sqrt{n+1}-\sqrt n\\ \\ \\ Ecua\c{\it t}ia\ \ devine:\\ \\ \sqrt2-1+\sqrt3-\sqrt2+\ ...\ +\sqrt{n+1}-\sqrt n=2019 \Rightarrow -1+\sqrt{n+1}=2019|_{+1}\Rightarrow\\ \\ \sqrt{n+1}=2020 \Rightarrow (\sqrt{n+1})^2=2020^2 \Rightarrow n+1=2020^2 \Rightarrow n=2020^2-1 \Rightarrow \\ \\ \Rightarrow n=(2020-1)(2020+1) \Rightarrow n=2019\cdot2021[/tex]
