Đáp án:
`A = 3/(1 xx 4) + 3/(4 xx 7) + ............... + 3/(97 xx 100)`
` = 1 - 1/4 + 1/4 - 1/7 + ............... + 1/97 - 1/100`
` = 1 - 1/100 = 99/100`
`B = 3/(1 xx 3) + 3/( 3 xx 5) + .................. + 3/( 99 xx 101)`
` = 3/2 xx ( 2/(1 xx 3) + 2/(3 xx 5) + ............. + 2/(99 xx 101))`
` = 3/2 xx ( 1 - 1/3 + 1/3 - 1/5 + ...............+ 1/99 - 1/101)`
` = 3/2 xx ( 1 - 1/101)`
` = 3/2 xx 100/101`
` =150/101`
`C = 1/( 1 xx 4) + 1/(4 xx 7) + ........ + 1/(52 xx 55)`
` = 1/3 xx ( 3/(1 xx 4) + 3/(4 xx 7) + .............. +3/(52 xx 55) )`
` = 1/3 xx ( 1 - 1/4 + 1/4 - 1/7 + ................ + 1/52 - 1/55)`
` = 1/3 xx ( 1 - 1/55)`
` = 1/3 xx 54/55`
` = 18/55`
`D = 10 xx ( 1/2 + 5/6 + 11/12 + 19/20 + .......... + 89/90)`
` = 10 xx ( 1 - 1/2 + 1 - 1/6 + 1 - 1/12 + .......... + 1 - 1/90)`
` = 10 xx [ ( 1 + 1 + 1 + ............ + 1) - (1/2 + 1/6 + 1/12 + ............... + 1/90)]`
` = 10 xx [ 9 - ( 1/( 1 xx 2) + 1/(2 xx 3) + 1/(3 xx 4) + ........... + 1/( 9 xx 10) ) ]`
` = 10 xx [ 9 - ( 1 - 1/10)]`
` = 10 xx [9 - 9/10]`
` = 10 xx 81/10`
` = 81`
