`a)` $\dfrac{1}{1.2}$ + $\dfrac{1}{2.3}$ + $\dfrac{1}{3.4}$ + .... + $\dfrac{1}{a(a+1)}$
` = 1/1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ... + 1/a - 1/(a+1)`
` = 1/1 + ( - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ... + 1/a ) - 1/(a+1) `
` = 1/1 + 0 - 1/(a+1) `
` =1/1 - 1/(a +1 ) = (a+1)/(a+1) - 1/(a+1) = a/(a+1)`
`b)` $\dfrac{2}{1.3}$ + $\dfrac{2}{3.5}$ + $\dfrac{2}{5.7}$ + ... + $\dfrac{2}{(2n+1 )(2n+3)}$
`= 1/1 - 1/3 + 1/3 - 1/5 + 1/5 - 1/7 + .... + 1/(2n+1) - 1/(2n+3)`
`= 1/1 + 0 - 1/(2n+3)`
`= 1/1 - 1/(2n+3) = (2n+3)/(2n+3) - 1/(2n+3) = (2n+2)/(2n+3) `
`c) 1/3 + 1/15 + 1/35 + 1/63 + 1/99`
`= 1/(1.3) + 1/(3.5) + 1/(5.7) + 1/(7.9) + 1/(9.11)`
Đặt `B` = ` 1/(1.3) + 1/(3.5) + 1/(5.7) + 1/(7.9) + 1/(9.11)`
`2B = 2/(1.3) + 2/(3.5) + 2/(5.7) + 2/(7.9) + 2/(9.11) `
`2B = 1/1 - 1/3 + 1/3 - 1/5 + .... + 1/9 - 1/11 `
`2B = 1/1 - 1/11 = 10/11`
`→ B = 10/11 : 2 = 10/11 . 1/2 = 5/11`
