giúp em với ạ em cần gấp lắm ạ

Đáp án:

$T=100$.

Giải thích các bước giải:

$\quad\ T=\dfrac{\dfrac1{99}+\dfrac2{98}+\dfrac3{97}\ +\,.\!.\!.+\ \dfrac{99}1}{\dfrac12+\dfrac13+\dfrac14\ +\,.\!.\!.+\ \dfrac1{100}}\\\Rightarrow T=\dfrac{\dfrac1{99}+\dfrac2{98}+\dfrac3{97}\ +\,.\!.\!.+\ 99}{\dfrac12+\dfrac13+\dfrac14\ +\,.\!.\!.+\ \dfrac1{100}}\\\Rightarrow T=\dfrac{\dfrac1{99}+\dfrac2{98}+\dfrac3{97}\ +\,.\!.\!.+\ \overbrace{1+1\ +\,.\!.\!.+\ 1}^{\large99}}{\dfrac12+\dfrac13+\dfrac14\ +\,.\!.\!.+\ \dfrac1{100}}\\\Rightarrow T=\dfrac{\left(\dfrac1{99}+1\right)\!+\!\left(\dfrac1{98}+1\right)\!+\!\left(\dfrac1{97}+1\right)+\,.\!.\!.+\ 1}{\dfrac12+\dfrac13+\dfrac14\ +\,.\!.\!.+\ \dfrac1{100}}\\\Rightarrow T=\dfrac{\dfrac{100}2+\dfrac{100}3+\dfrac{100}4\ +\,.\!.\!.+\ \dfrac{100}{100}}{\dfrac12+\dfrac13+\dfrac14\ +\,.\!.\!.+\ \dfrac1{100}}\\\Rightarrow T=\dfrac{100.\!\;\!\!\left(\dfrac12+\dfrac13+\dfrac14\ +\,.\!.\!.+\ \dfrac1{100}\right)}{\dfrac12+\dfrac13+\dfrac14\ +\,.\!.\!.+\ \dfrac1{100}}$

$\Rightarrow T=100$.