2*3*4*5分之1+3*4*5*6分之1+4*5*6*7分之1•••••+13*14*15*16分之1等于几

题目

2*3*4*5分之1+3*4*5*6分之1+4*5*6*7分之1•••••+13*14*15*16分之1等于几

答案

2*3*4*5分之1+3*4*5*6分之1+4*5*6*7分之1•••••+13*14*15*16分之1
=1/2*[1/(2*5)-1/(3*4)]+1/2*[1/(3*6)-1/(4*5)]+1/2*[1/(4*7)-1/(5*6)]+……+1/2*[1/(13*16)-1/(14*15)]
=1/2*[1/3*(1/2-1/5)-(1/3-1/4)]+1/2*[1/3*(1/3-1/6)-(1/4-1/5)]+1/2*[1/3*(1/4-1/7)-(1/5-1/6)]+……+1/2*[1/3*(1/13-1/16)-(1/14-1/15)]
=1/6*(1/2-1/5+1/3-1/6+1/4-1/7+……+1/13-1/16)-1/2*(1/3-1/4+1/4-1/5+1/5-1/6+……+1/14-1/15)
=1/6*(1/2+1/3+1/4-1/14-1/15-1/16)-1/2*(1/3-1/15)
=1/6*(3/7+4/15+3/16)-2/15
=1/14+2/45+1/32-2/15
=1/14+1/32-4/45
=139/10080