小六數學1/3+1/6+1/10+1/15+1/21=

1/3+1/6+1/10+1/15+1/21+1/28+1/36+1/45+1/55=?
Sol
1/6+1/12+1/20+1/30+1/42+1/56+1/72+1/90+1/110
=1/(2*3)+1/(3*4)+1/(4*5)+1/(5*6)+1/(6*7)+1/(7*8)+1/(8*9)+1/(9*10)+1/(10*11)
=(3－2)/(2*3)+(4－3)/(3*4)+(5－4)/(4*5)+(6－5)/(5*6)+(7－6)/(6*7)
+(8－7)/(7*8)+(9－8)/(8*9)+(10－9)/(9*10)+(11－10)/(10*11)
=[3/(2*3)－2/(2*3)]+[4/(3*4)－3/(3*4)]+[5/(4*5)－4/(4*5)]+[7/(5*6)－5/(5*6)]
+[8/(7*8)－7/(7*8)]+[9/(9*10)－10/(9*10)]+[11/(10*11)－10/(10*11)
=(1/2－1/3)+(1/3－1/4)+(1/4－1/5)+(1/5－1/6)+(1/6－1/7)+(1/7－1/8)
+(1/8－1/9)+(1/9－1/10)+(1/10－11/1)
=1/2－1/11
=9/22
So
1/3+1/6+1/10+1/15+1/21+1/28+1/36+1/45+1/55=9/11

1/3 + 1/6 + 1/10 + 1/15 + 1/21 + 1/28 + 1/36 + 1/45 + 1/55

= 2/6 + 2/12 + 2/20 + 2/30 + 2/42 + 2/56 + 2/72 + 2/90 + 2/110

= (2/2 - 2/3) + (2/3 - 2/4) + (2/4 - 2/5) + (2/5 - 2/6) + (2/6 - 2/7) + (2/7 - 2/8) + (2/8 - 2/9) + (2/9 - 2/10) + (2/10 - 2/11)

= 2/2 - 2/11

= 9/11

William高手
給你一個讚
多項相消才是正法
若一百項以上不是闊分算累塌

原式=1/1 + 1/3 + 1/6 + 1/10 + 1/15 + 1/21 + ….+ 1/55 – 1
=1/1 + 1/(1+2) + 1/(1+2+3) + …+ 1/(1+2+3+…+10) – 1
=1/[(1*2)/2] + 1/[(2*3)/2] + 1/[(3*4)/2] + …+ 1/[(10*11)/2] – 1
=2[1/(1*2) + 1/(2*3) + 1/(3*4) + …+ 1/(10*11)] – 1
=2[(1/1)–(1/2)+(1/2)–(1/3)+(1/3)–(1/4)….+(1/10)–(1/11)] – 1

2013-03-25 11:09:18 補充：
=2(1/1 – 1/11) – 1
=2*(10/11) – 1
=20/11 – 11/11
= 9/11

1/3 + 1/6 + 1/10 + 1/15 + 1/21 + 1/28 + 1/36 + 1/45 + 1/55

= (1/3 + 1/6) + (1/10 + 1/15) + (1/21 + 1/28) + (1/36 + 1/45) + 1/55

= (2/6 + 1/6) + (3/30 + 2/30) + (4/84 + 3/84) + (5/180 + 4/180) + 1/55

= 3/6 + 5/30 + 7/84 + 9/180 + 1/55

= (3/6 + 1/6) + (1/12 + 1/20) + 1/55

2013-03-25 10:23:58 補充：
= 4/6 + (5/60 + 3/60) + 1/55

= (2/3 + 8/60) + 1/55

= (10/15 + 2/15) + 1/55

= 12/15 + 1/55

= 4/5 + 1/55

= 44/55 + 1/55

= 45/55

= 9/11 ...... 答案

先求3.6.10.15.21.28.36.45.55的最小公倍數 13860 通分後再加總
(4620+2310+1386+924+660+495+385+308+252)/13860
=11340/13860 = 9/11