5/2x3x4+7/3x4x5+...+19/9x10x11
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5/(2*3*4) = 1/(2*4) + 1/(3*4) = 1/2 *( 1/2 - 1/4) + ( 1/3 - 1/4)
7/(3*4*5) = 1/(3*5) + 1/(4*5) = 1/2 *(1/3 - 1/5) + (1/4 - 1/5)
9/(4*5*6) = 1/(4*6) + 1/(5*6) = 1/2 *(1/4 - 1/6) + ( 1/5 - 1/6)
11/(5*6*7) = 1/(5*7) + 1/(6*7) = 1/2*(1/5 - 1/7) + (1/6 - 1/7)
13/(6*7*8) = 1/(6*8) + 1/(7*8) = 1/2*(1/6 - 1/8) + (1/7 - 1/8)
15/(7*8*9) = 1/(7*9) + 1/(8*9) = 1/2*(1/7 - 1/9) + (1/8 - 1/9)
17/(8*9*10) = 1/(8*10) + 1/(9*10) = 1/2*(1/8 - 1/10) + (1/9 - 1/10)
19/(9*10*11) = 1/(9*11) + 1/(10*11) = 1/2*(1/9 - 1/11) + (1/10 - 1/11)
各式相加得
原式 = 1/2 *( 1/2 + 1/3 - 1/10 - 1/11) + (1/3 - 1/11)
= 1/4 +1/6 - 1/20 - 1/22 + 1/3 - 1/11
= (1/4 + 1/6 + 1/3) - 1/20 - (1/22 + 1/11)
= 3/4 - 1/20 - 3/22
= 7/10 - 3/22
= 31/55
7/(3*4*5) = 1/(3*5) + 1/(4*5) = 1/2 *(1/3 - 1/5) + (1/4 - 1/5)
9/(4*5*6) = 1/(4*6) + 1/(5*6) = 1/2 *(1/4 - 1/6) + ( 1/5 - 1/6)
11/(5*6*7) = 1/(5*7) + 1/(6*7) = 1/2*(1/5 - 1/7) + (1/6 - 1/7)
13/(6*7*8) = 1/(6*8) + 1/(7*8) = 1/2*(1/6 - 1/8) + (1/7 - 1/8)
15/(7*8*9) = 1/(7*9) + 1/(8*9) = 1/2*(1/7 - 1/9) + (1/8 - 1/9)
17/(8*9*10) = 1/(8*10) + 1/(9*10) = 1/2*(1/8 - 1/10) + (1/9 - 1/10)
19/(9*10*11) = 1/(9*11) + 1/(10*11) = 1/2*(1/9 - 1/11) + (1/10 - 1/11)
各式相加得
原式 = 1/2 *( 1/2 + 1/3 - 1/10 - 1/11) + (1/3 - 1/11)
= 1/4 +1/6 - 1/20 - 1/22 + 1/3 - 1/11
= (1/4 + 1/6 + 1/3) - 1/20 - (1/22 + 1/11)
= 3/4 - 1/20 - 3/22
= 7/10 - 3/22
= 31/55
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