Question

Se considera numarul a=1+1 supra 1+2 +1 supra 1+2+3+.......+ 1 supra 1+2+....+200 aratati ca numarul 2,01 ori a este patratul unui numar natural .

Answer 1

[tex]a=1+\dfrac{1}{1+2}+\dfrac{1}{1+2+3}+\ldots+\dfrac{1}{1+2+\ldots+200}\\ \\ a=1+\dfrac{1}{\frac{2\cdot 3}{2}}+\dfrac{1}{\frac{3\cdot 4}{2}}+\ldots +\dfrac{1}{\frac{200\cdot 201}{2}}\\ \\ a=1+\dfrac{2}{2\cdot 3}+\dfrac{2}{3\cdot 4}+\ldots+\dfrac{2}{200\cdot 201}\\ a=2\left(\dfrac{1}{1\cdot 2}+\dfrac{1}{2\cdot 3}+\ldots +\dfrac{1}{200\cdot 201}\right)\\ \\ a=2\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\ldots +\dfrac{1}{200}-\dfrac{1}{201}\right)\\ \\ a=2\cdot \left(1-\dfrac{1}{201}\right) [/tex]

[tex]a=2\cdot \dfrac{200}{201}\\ \\ 2,01\cdot a=\dfrac{201}{100} \cdot 2\cdot \dfrac{200}{201}=4=2^2[/tex]