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1/1*2*3*4+1/2*3*4*5+1/3*4*5*6+••••••+1/17*18*19*20=
题目详情
1/1*2*3*4+1/2*3*4*5+1/3*4*5*6+••••••+1/17*18*19*20=_______
▼优质解答
答案和解析
由于1/[n(n+1)(n+2)(n+3)]
=(1/3)[(n+3)-n]/[n(n+1)(n+2)(n+3)]
=(1/3)[1/n(n+1)(n+2)-1/(n+1)(n+2)(n+3)]
故原式=(1/3)(1/1*2*3-1/2*3*4+1/2*3*4-···+1/17*18*19-1/18*19*20)
=(1/3)(1/1*2*3-1/18*19*20)
=1139/20520.
=(1/3)[(n+3)-n]/[n(n+1)(n+2)(n+3)]
=(1/3)[1/n(n+1)(n+2)-1/(n+1)(n+2)(n+3)]
故原式=(1/3)(1/1*2*3-1/2*3*4+1/2*3*4-···+1/17*18*19-1/18*19*20)
=(1/3)(1/1*2*3-1/18*19*20)
=1139/20520.
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