(2/5*7*9)+(2/7*9*11)+....

Question

(2/5*7*9)+(2/7*9*11)+(2/9*11*13)+(2/11*13*15)有比較快的拆解方式嗎??

2
---------- 題目是這樣~不好意思~沒打清楚!
5*7*9

Answer 1

2/(5×7×9)+2/(7×9×11)+2/(9×11×13)+2/(11×13×15)=(1/2)[4/(5×7×9)+4/(7×9×11)+4/(9×11×13)+4/(11×13×15)]=(1/2)[1/(5×7)-1/(7×9)+1/(7×9)-1/(9×11)+1/(9×11)-1/(11×13)+1/(11×13)-1/(13×15)]=(1/2)[1/(5×7)-1/(13×15)]=(1/2)[39/(5×7×39)-7/(13×15×7)]=(1/2)[32/(5×7×39)]=16/1365

Answer 2

利用2/[(x-2)*x*(x+2)] = 0.5/[(x-2)*x] - 0.5/[x*(x+2)]例如2/(5*7*9)=0.5/(5*7) - 0.5/(7*9)(2/5*7*9)+(2/7*9*11)+(2/9*11*13)+(2/11*13*15)=0.5/(5*7)-0.5/(7*9)+0.5/(7*9)-0.5/(9*11)+0.5/(9*11)-0.5/(11*13)+0.5/(11*13)-0.5/(13*15)=0.5/(5*7)-0.5/(13*15)=1/70-1/390=32/2730=16/1365

Answer 3

這是我目前推出來的
1/X - 1/(X+2) = 2 / X(X+2)
1/7 - 1/9 = 2 / 7*9
前兩項分母都有7和9 後兩項分母都有11和13 化成
1/5(1/7-1/9)+1/11(1/7-1/9)+1/9(1/11-1/13)+1/15(1/11-1/13)
=(1/5+1/11)(1/7-1/9)+(1/9+1/15)(1/11-1/13)
=16/55*2/63+8/45*2/143
=16/1365
好像沒有比較好算 XD

Answer 4

可以將除轉換成為乘法方便邏輯運算統一化,例如:(0.4*7*9)+(0.29*9*11)+(0.2*11*13)+(0.18*13*15) ==>(0.4*7*9)+(0.3*9*11)+(0.2*11*13)+(0.2*13*15){小數點1位).
缺點就是答案可能會有誤差.( + - 3左右).
1.你的題目計算約為120.716
2.轉化成乘法約為117.61
3.轉化為容易相乘的乘法122.5