Question

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1/2+1/2²+1/2³+...............+1/2¹⁰

1/1.2.3+1/2.3.4+.........+1/98.99.100

Answer 1

\(B=\dfrac{1}{1.2.3}+\dfrac{1}{2.3.4}+..+\dfrac{1}{98.99.100}\)

\(\dfrac{1}{1.2}-\dfrac{2}{2.3}=\dfrac{2}{1.2.3};...;\dfrac{1}{98.99}-\dfrac{1}{99.100}=\dfrac{2}{98.99.100}\)

=> 2B = \(\dfrac{2}{1.2.3}+\dfrac{2}{2.3.4}+.....+\dfrac{2}{98.99.100}\)

\(=\dfrac{1}{1.2}-\dfrac{1}{2.3}+\dfrac{1}{2.3}-\dfrac{1}{3.4}+..+\dfrac{1}{98.99}-\dfrac{1}{99.100}\)

=> 2B = \(\dfrac{1}{2}-\dfrac{1}{99.100}=\dfrac{4949}{9900}\)

=> B = \(\dfrac{4949}{19800}\)