1 + 31/6 + 51/12 + 71/20 + 91/30 + 111/42

= 1 + (30+1)/6 + (48+3)/12 + (60+11)/20 + (90+1)/30 + (84+27)/42

= 1 + 5+ 1/6 + 4+ 1/4 + 3+ 11/20 + 3+ 1/30 + 2+ 9/14

= 1 + 5 + 4 + 3 + 3 + 2 + 10/60 + 15/60 + 33/60 + 2/60 + 9/14

= 18 + 25/60 + 35/60 + 9/14

= 18 + 60/60 + 9/14

= 19 + 9/14

1/(1X4) + 1/(4X7) + 1/(7X10) + …… + 1/(73X76)

= 3/3 X [ 1/(1X4) + 1/(4X7) + 1/(7X10) + …… + 1/(73X76) ]

= (1/3) X [ 3/(1X4) + 3/(4X7) + 3/(7X10) + …… + 3/(73X76) ]

= (1/3) X [ (4-1)/(1X4) + (7-4)/(4X7) + (10-7)/(7X10) + ……+ (76-73)/(73X76) ]

= (1/3) X [ 4/4 - 1/4 + 1/4 - 1/7 + 1/7 - 1/10 + 1/10 + …… - 1/73 + 1/73 - 1/76 ]

= (1/3) X ( 1 - 1/76 )

= 1/3 X 75/76

= 25/76

8787 8787 X 777 7777 / 1010101 / 111 1111

= 87878787 / 1010101 X 777 7777 / 111 1111

= 87 X 7

= ( 90 - 3 ) X 7

= 630 - 21

= 609

99999 X 22222 + 33333 X 33334

= 33333 X 66666 + 33333 X 33334

= 33333 X ( 66666 + 33333 + 1 )

= 33333 X ( 99999 + 1 )

= 33333 X 100000

= 33 3330 0000