Question

($\frac{1}{1.2}$ + $\frac{1}{2.3}$ + $\frac{1}{3.4}$ + ... + $\frac{1}{2011. 2012}$ ) x = 2011

Answer 1

\(\Leftrightarrow x\cdot\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{2011}-\dfrac{1}{2012}\right)=2011\)

\(\Leftrightarrow x\cdot\dfrac{2011}{2012}=2011\)

hay x=2012

Answer 2

\(\left(\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{2011.2012}\right)x=2011\)

\(\left(\dfrac{1}{1}-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{2011}-\dfrac{1}{2012}\right)x=2011\)

\(\left(\dfrac{1}{1}-\dfrac{1}{2012}\right)x=2011\)

\(\dfrac{2011}{2012}x=2011\)
\(x=2012\)

Answer 3

`(1/[1.2]+1/[2.3]+1/[3.4]+....+1/[2011.2012])x=2011`

`(1-1/2+1/2-1/3+1/3-1/4+.....+1/2011-1/2012)x=2011`

`(1-1/2012)x=2011`

`2011/2012x=2011`

`x=2011:2011/2012`

`x=2012`

Answer 4

\(\left(\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot4}+...+\dfrac{1}{2011\cdot2012}\right)\cdot x=2011\)

\(\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{2011}+\dfrac{1}{2011}-\dfrac{1}{2012}\right)\cdot x=2011\)

\(\left(1-\dfrac{1}{2012}\right)\cdot x=2011\)

\(\dfrac{2011}{2012}\cdot x=2011\)

 \(x=2011:\dfrac{2011}{2012}\)

 \(x=2012\)