Cho Q=1/2x2+1/3x3+1/4x4+...+1/200x200.so sánh Q với 3/4

$\texttt{Đáp án:}$

`Q>\frac{3}{4}`

$\texttt{Giải thích các bước giải:}$

$Q= \dfrac{1}{2\times 2}+ \dfrac{1}{3\times 3}+ \dfrac{1}{4\times 4}+...+ \dfrac{1}{2020\times 2020}$

$\texttt{Ta thấy:}$

$\dfrac{1}{2\times 2}> \dfrac{1}{2\times 3}$

$\dfrac{1}{3\times 3}> \dfrac{1}{3\times 4}$

$\dfrac{1}{4\times 4}> \dfrac{1}{4\times 5}$

$\dfrac{1}{2020\times 2020}> \dfrac{1}{2020\times 2021}$

$\Rightarrow Q> \dfrac{1}{2\times 3}+\dfrac{1}{3\times 4}+\dfrac{1}{4\times 5}+...+\dfrac{1}{2020\times 2021}$

$\Rightarrow Q> \dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+.....+\dfrac{1}{2020}-\dfrac{1}{2021}$

$\Rightarrow Q> 1-\dfrac{1}{2021}$

$\Rightarrow Q<\dfrac{2020}{2021}$

$\texttt{Mà:}$ $\dfrac{2020}{2021}>\dfrac{3}{4}$

$\Rightarrow Q>\dfrac{3}{4}$

`=>` $\texttt{Vậy:}$ `Q = 1/2xx2 + 1/3xx3 + 1/4xx4 + ... +1/2020xx2020> 3/4`