1/1.3+1/3.5+1/5.7+...+1/(2x-1)(2+1)=49/99
1 câu trả lời
Đáp án:
$x=49.$
Giải thích các bước giải:
$\dfrac{1}{1.3}+\dfrac{1}{3.5}+\dfrac{1}{5.7}+\dots+\dfrac{1}{(2x-1)(2x+1)}=\dfrac{49}{99}\\ \Leftrightarrow \dfrac{2}{1.3}+\dfrac{2}{3.5}+\dfrac{2}{5.7}+\dots+\dfrac{2}{(2x-1)(2x+1)}=\dfrac{98}{99}\\ \Leftrightarrow \dfrac{3-1}{1.3}+\dfrac{5-3}{3.5}+\dfrac{7-5}{5.7}+\dots+\dfrac{2x+1-(2x-1)}{(2x-1)(2x+1)}=\dfrac{98}{99}\\ \Leftrightarrow \dfrac{3}{1.3}-\dfrac{1}{1.3}+\dfrac{5}{3.5}-\dfrac{3}{3.5}+\dfrac{7}{5.7}-\dfrac{5}{5.7}+\dots+\dfrac{2x+1}{(2x-1)(2x+1)}-\dfrac{2x-1}{(2x-1)(2x+1)}=\dfrac{98}{99}\\ \Leftrightarrow 1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+\dots+\dfrac{1}{2x-1}-\dfrac{1}{2x+1}=\dfrac{98}{99}\\ \Leftrightarrow 1-\dfrac{1}{2x+1}=\dfrac{98}{99}\\ \Leftrightarrow \dfrac{2x+1-1}{2x+1}=\dfrac{98}{99}\\ \Leftrightarrow \dfrac{2x}{2x+1}=\dfrac{98}{99}\\ \Leftrightarrow 99.2x=98(2x+1)\\ \Leftrightarrow 198x=196x+98\\ \Leftrightarrow 2x=98\\ \Leftrightarrow x=49.$