Question

1*2*3/1+2*3*4/1+3*4*5/1+``````+18*19*20/1=

Answer 1

1/(n-1)(n+1)n=1/2[1/n(n-1)-1/n(n+1)]
所以原式=1/2[1/(2*1)-1/(2*3)]+1/2[1/(3*2)-1/(3*4)]+......+1/2[1/(19*18)-1/(19*20)]
=1/2[1/(2*1)-1/(2*3)+1/(3*2)-1/(3*4)+......+1/(19*18)-1/(19*20)]
=1/2[1/2-1/(19*20)]
=189/760

这是我在静心思考后得出的结论，
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Answer 2

1/(1*2*3)+1/(2*3*4)+1/(3*4*5)+...+1/(18*19*20)

an = 1/[n(n+1)(n+2)]
= 1/2{1/[n(n+1)] - 1/[(n+1)(n+2)]}
Sn = (1/2)( 1/2 -1/[(n+1)(n+2)] }
1/(1*2*3)+1/(2*3*4)+1/(3*4*5)+...+1/(18*19*20)
=S18
=(1/2)( 1/2 -1/[19.20] }
= 189/760