Question

Tìm số tự nhiên x, biết rằng:

\(\dfrac{1}{10}+\dfrac{1}{15}+\dfrac{1}{21}+...+\dfrac{2}{x.\left(x+1\right)}=\dfrac{2010}{2012}\)

Answer 1

https://hoc24.vn/question/246430.html

Answer 2

\(\dfrac{1}{10}+\dfrac{1}{15}+\dfrac{1}{21}+...+\dfrac{2}{x.\left(x+1\right)}=\dfrac{2010}{2012}\)

\(\dfrac{2}{20}+\dfrac{2}{30}+\dfrac{2}{42}+...+\dfrac{2}{x.\left(x+1\right)}=\dfrac{2010}{2012}\)

\(2\left(\dfrac{1}{20}+\dfrac{1}{30}+\dfrac{1}{42}+...+\dfrac{1}{x.\left(x+1\right)}\right)=\dfrac{2010}{2012}\)

\(2\left(\dfrac{1}{4.5}+\dfrac{1}{5.6}+\dfrac{1}{6.7}+...+\dfrac{1}{x.\left(x+1\right)}\right)=\dfrac{2010}{2012}\)

\(2\left(\dfrac{1}{4}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{7}+...+\dfrac{1}{x}-\dfrac{1}{\left(x+1\right)}\right)=\dfrac{2010}{2012}\)

\(2\left(\dfrac{1}{4}-\dfrac{1}{\left(x+1\right)}\right)=\dfrac{2010}{2012}\)

\(\dfrac{1}{4}-\dfrac{1}{\left(x+1\right)}=\dfrac{2010}{2012}:2\)

\(\dfrac{1}{4}-\dfrac{1}{\left(x+1\right)}=\dfrac{1005}{2012}\)

\(\dfrac{1}{\left(x+1\right)}=\dfrac{1}{4}-\dfrac{1005}{2012}\)

\(\dfrac{1}{\left(x+1\right)}=\dfrac{-251}{1006}\)

\(\Rightarrow1:\left(x+1\right)=\dfrac{-251}{1006}\)

\(\left(x+1\right)=1:\dfrac{-251}{1006}\)

\(x+1=\dfrac{-1006}{251}\)

\(x=\dfrac{-1006}{251}-1\)

\(x=\dfrac{-1257}{251}\)

Vì \(x\in N\) nên \(x=\varnothing\) (không có giá trị nào của x thoả mãn)

Answer 3

Ta có :

\(\dfrac{1}{10}+\dfrac{1}{15}+\dfrac{1}{21}+.........+\dfrac{2}{x\left(x+1\right)}=\dfrac{2010}{2012}\)

\(\dfrac{2}{20}+\dfrac{2}{30}+\dfrac{2}{42}+..........+\dfrac{2}{x\left(x+1\right)}=\dfrac{2010}{2012}\)

\(\dfrac{2}{4.5}+\dfrac{2}{5.6}+\dfrac{2}{7.8}+.......+\dfrac{2}{x\left(x+1\right)}=\dfrac{2010}{2012}\)

\(2\left(\dfrac{1}{4.5}+\dfrac{1}{5.6}+\dfrac{1}{7.8}+...........+\dfrac{1}{x\left(x+1\right)}\right)=\dfrac{2010}{2012}\)

\(2\left(\dfrac{1}{4}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{6}+.........+\dfrac{1}{x}-\dfrac{1}{x+1}\right)=\dfrac{2010}{2012}\)

\(2\left(\dfrac{1}{4}-\dfrac{1}{x+1}\right)=\dfrac{2010}{2012}\)

\(\dfrac{1}{4}-\dfrac{1}{x+1}=\dfrac{2010}{2012}:2\)

\(\dfrac{1}{4}-\dfrac{1}{x+1}=\dfrac{1005}{2012}\)

\(\dfrac{1}{x+1}=\dfrac{1}{4}-\dfrac{1005}{2012}\)

\(\dfrac{1}{x+1}=\dfrac{-215}{1006}\)

\(\Rightarrow1.1006=\left(x+1\right).\left(-215\right)\)

\(1006=\left(x+1\right).\left(-215\right)\)

\(x+1=1006:\left(-215\right)\)

\(x+1=\dfrac{-1006}{215}\)

\(x=\dfrac{-1006}{215}-1\)

\(x=\dfrac{-1221}{215}\)(ko thỏa mãn \(x\in N\))

Vậy ko tìm dc giá trị của x thỏa mãn theo yêu cầu

~ Học tốt ~