$E = \dfrac{1}{1.3} + \dfrac{1}{3.5} + \dfrac{1}{5.7} + .. . + \dfrac{1}{19.21}$
$B = \dfrac{1}{9.10} – \ dfrac{8.9} – \dfrac{1}{7.8} – … – \dfrac{1}{2.3} – \dfrac{1}{1.2}$
$E = \dfrac{1}{1.3} + \dfrac{1}{3.5} + \dfrac{1}{5.7} + .. . + \dfrac{1}{19.21}$ $B = \dfrac{1}{9.10} – \ dfrac{8.9} – \dfrac{1}{7.8} – … – \dfra
By Amara
`E =1/(1*3)+1/(3*5)+1/(5*7)+…+1/(19*21)`
`⇔2E =2/(1*3)+2/(3*5)+2/(5*7)+…+2/(19*21)`
`=1-1/3+1/3-1/5+1/5-1/7+1/7-1/9+…+1/19-1/21`
`=1-1/21=20/21`
`⇒E = 10/21`
`B=1/(9*10)-1/(8*9)-1/(7*8)-…-1/(1*2)`
`=-1/(1*2)-1/(2*3)-1/(3*4)-…-1/(8*9)+1/(9*10)`
`=-(1/(1*2)+1/(2*3)+1/(3*4)+…+1/(8*9))+1/(9*10)`
`=1/(9*10)-(1/(1*2)+1/(2*3)+1/(3*4)+…+1/(8*9))`
`=1/(9*10)-(1-1/2+1/2-1/3+1/3-1/4+…+1/8-1/9)`
`=1/(9*10)-(1-1/9)=1/(9*10)-8/9`
`=1/90-8/9=1/90-80/90=-79/90`
`E = \frac{1}{1.3} + \frac{1}{3.5} + 1/{5.7} + … + 1/{19.21}`
$2E = \dfrac{2}{1.3} + \dfrac{2}{3.5} + \dfrac{2}{5.7} + … + \dfrac{1}{19.21}$
`2E = 1/1 – 1/3 + 1/3 – 1/5 + 1/5 – 1/7 + … + 1/{19} – 1/{21}`
`2E = 1 – 1/21`
`2E = 20/21`
`2E = 10/21`
`B = 1/{9.10} – 1/{8.9} – 1/{7.8} – … – 1/{2.3} – 1/{1.2}`
`B = 1/{9.10} – (1/{1.2} + 1/{2.3} + … + 1/{7.8} + 1/{8.9})`
`B = 1/{90} – (1/1 – 1/2 + 1/2 – 1/3 + … + 1/7 – 1/8 + 1/8 – 1/9)`
`B = 1/{90} – 1 + 1/9`
`B = -79/90`.