Question

H = 6/15 x 3 - 14/15 x 27 + 22/27 x 39 - 30/39 x 51 +....+      70/99 x 111

Answer 1

\(H=\dfrac{6}{3\cdot15}-\dfrac{14}{15\cdot27}+\dfrac{22}{27\cdot39}-...+\dfrac{70}{99\cdot111}\)

\(=\dfrac{1}{3}\left(\dfrac{18}{3\cdot15}-\dfrac{42}{15\cdot27}+\dfrac{66}{27\cdot39}-...+\dfrac{210}{99\cdot111}\right)\)

\(=\dfrac{1}{3}\left(\dfrac{1}{3}+\dfrac{1}{15}-\dfrac{1}{15}-\dfrac{1}{27}+...+\dfrac{1}{99}+\dfrac{1}{111}\right)\)

\(=\dfrac{1}{3}\left(\dfrac{1}{3}+\dfrac{1}{111}\right)=\dfrac{1}{3}\cdot\dfrac{37+1}{111}=\dfrac{38}{333}\)