tìm x B, 1/3+1/6+1/10+...+2/x.(x+1)= 499/1000

$ĐK:x\neq 0;x\neq -1$

Xét vế trái:

$\dfrac{1}{3}+\dfrac{1}{6}+\dfrac{1}{10}+...+\dfrac{2}{x(x+1)}\\ =2.(\dfrac{1}{6}+\dfrac{1}{12}+\dfrac{1}{20}+...+\dfrac{1}{x(x+1)})\\ =2.(\dfrac{1}{2.3}+\dfrac{1}{3.4}+\dfrac{1}{4.5}+...+\dfrac{1}{x(x+1)}\\ =2.(\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+...+\dfrac{1}{x}-\dfrac{1}{x+1})\\ =2.(\dfrac{1}{2}-\dfrac{1}{x+1})$

Thay vào PT:

$\Rightarrow 2.(\dfrac{1}{2}-\dfrac{1}{x+1})=\dfrac{499}{1000}\\ \Rightarrow \dfrac{1}{2}-\dfrac{1}{x+1}=\dfrac{499}{2000}\\ \Rightarrow \dfrac{1}{x+1}=\dfrac{1}{2}-\dfrac{499}{2000}=\dfrac{501}{2000}\\ \Rightarrow 501x+501=2000\\ \Rightarrow x=\dfrac{1499}{501}(tm)$

Vậy $x=\dfrac{1499}{501}$

`text{Đáp án + Giải thích các bước giải :}`

Điều kiện `: x \ne 0; x \ne -1 .`

`1/3 + 1/6 + 1/10 + ... + 2/(x.(x+1)) = 499/1000`

 `2/6 + 2/12 + 2/20 + ... + 2/(x.(x+1)) = 499/1000`

 `2/(2.3) + 2/(3.4) + 2/(4.5) + ... + 2/(x.(x+1)) = 499/1000`

`(2/(2.3) + 2/(3.4) + 2/(4.5) + ... + 2/(x.(x+1))) . 2/4 = 499/1000 . 2/4`

`1/(2.3) + 1/(3.4) + 1/(4.5) + ... + 1/(x.(x+1)) = 499/2000`

`1/2 - 1/3 + 1/3 - 1/4 + 1/4 - 1/5 + ... + 1/x - 1/(x+1) = 499/2000`

`1/2 - 1/(x+1) = 499/2000`

`1/(x+1) = 1/2 - 499/2000`

`1/(x+1) = 501/2000`

`x + 1  . 501 = 2000`

`x + 1 = 2000 : 501`

`x + 1 = 2000/501`

`x = 2000/501 - 1`

`x = 1499/501 .`

Vậy `x = 1501/499 .`

`text{#Muối}`