A = 1/1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 +...+ 1/2011 - 1/2012

A = 1 - 1/2012

A = 2011/2012

B = 1/2 - 1/4 + 1/4 - 1/6 + 1/6 - 1/8 +...+ 1/2010 - 1/2012

B = 1/2 - 1/2012

B = 1005/2012

a) \(A=\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot4}+...+\dfrac{1}{2011\cdot2012}\)

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

\(A=1-\dfrac{1}{2012}\)

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

b) \(B=\dfrac{1}{2\cdot4}+\dfrac{1}{4\cdot6}+\dfrac{1}{6\cdot8}+...+\dfrac{1}{2010\cdot2012}\)

\(B=\dfrac{1}{2}\cdot\left(\dfrac{2}{2\cdot4}+\dfrac{2}{4\cdot6}+\dfrac{2}{6\cdot8}+...+\dfrac{2}{2010\cdot2012}\right)\)

\(B=\dfrac{1}{2}\cdot\left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{8}+...+\dfrac{1}{2010}-\dfrac{1}{2012}\right)\)

\(B=\dfrac{1}{2}\cdot\left(\dfrac{1}{2}-\dfrac{1}{2012}\right)\)

\(B=\dfrac{1}{2}\cdot\dfrac{1005}{2012}\)

\(B=\dfrac{1005}{4024}\)

\(\dfrac{2}{2.4}+\dfrac{2}{4.6}+................+\dfrac{2}{50.52}\)

\(=\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+..............+\dfrac{1}{50}-\dfrac{1}{52}\)

\(=\dfrac{1}{2}-\dfrac{1}{52}=\dfrac{25}{52}\)

\(\dfrac{2}{2.4}+\dfrac{2}{4.6}+.....+\dfrac{2}{50.52}\)

\(=\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+.....+\dfrac{1}{50}-\dfrac{1}{52}\)

\(=\dfrac{1}{2}-\dfrac{1}{52}=\dfrac{25}{52}\)

\(\dfrac{2}{2.4}+\dfrac{2}{4.6}+...+\dfrac{2}{x.\left(x+2\right)}=\dfrac{4}{9}\)

=\(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+...+\dfrac{1}{x}-\dfrac{1}{x-2}\)

=\(\dfrac{1}{2}-\dfrac{1}{x+2}\)=\(\dfrac{4}{9}\)

=>\(\dfrac{1}{x+2}=\dfrac{1}{2}-\dfrac{4}{9}=\dfrac{1}{18}\)

=>\(x+2=18\)

=>x=16

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