Salut,
Produsul din enunț se poate scrie așa:
[tex]P=\prod\limits_{k=2}^{100}\left(1-\dfrac{1}{k^2}\right)=\prod\limits_{k=2}^{100}\dfrac{k^2-1}{k^2}=\prod\limits_{k=2}^{100}\left[\dfrac{(k-1)(k+1)}{k^2}\right]=\prod\limits_{k=2}^{100}\left(\dfrac{k-1}{k}\cdot\dfrac{k+1}{k}\right)=\\\\=\left(\dfrac{1}{2}\cdot\dfrac{\bcancel3}{\bcancel2}\right)\cdot\left(\dfrac{\cancel2}{\cancel3}\cdot\dfrac{\cancel4}{\cancel3}\right)\cdot\left(\dfrac{\cancel3}{\cancel4}\cdot\dfrac{\cancel5}{\cancel4}\right)\ldots\left(\dfrac{\cancel{98}}{\cancel{99}}\cdot\dfrac{\cancel{100}}{\cancel{99}}\right)\cdot\left(\dfrac{\cancel{99}}{\cancel{100}}\cdot\dfrac{101}{100}\right)=\dfrac{101}{200}.[/tex]
Green eyes.
