Question

1/1*2*3*4+1/2*3*4*5+1/3*4*5*6+-----+1/17*18*19*20

Answer 1

1/n(n+1)(n+2)(n+3)
=1/(n^2+3n)(n^2+3n+2)
=1/2[1/(n^2+3n)-1/(n^2+3n+2)]
=1/2[1/n(n+3)-1/(n+1)(n+2)]
1/1*2*3*4+1/2*3*4*5+1/3*4*5*6+-----+1/17*18*19*20
=1/2[1/1*4-1/2*3+1/2*5-1/3*4+1/3*6-1/4*5+-----+1/17*20-1/18*19]
=1/2[1/1*4+1/2*5+1/3*6+-----+1/17*20-1/2*3-1/3*4-1/4*5-.-1/18*19]
=1/2[1/3(3/1*4+3/2*5+3/3*6+-----+3/17*20)-(1/2*3+1/3*4+1/4*5+.+1/18*19)]
=1/2[1/3(1-1/4+1/2-1/5+1/3-1/6+-----+1/17-1/20)-(1/2-1/3+1/3-1/4+1/4-1/5+.+1/18-1/19)]
=1/2[1/3(1-1/4+1/2-1/5+1/3-1/6+1/4-1/7+1/5-1/8+1/6-1/9+1/7-1/10+1/8-1/11+1/9-1/12+1/10-1/13+1/11-1/14+1/12-1/15+1/13-1/16+1/14-1/17+1/15-1/18+1/16-1/19+1/17-1/20)-(1/2-1/19)]
=1/2[1/3(1+1/2+1/3+1/5-1/17-1/18-1/19-1/20)-(1/2-1/19)]
=1/2[1/3+1/6+1/9+1/15-1/51-1/54-1/57-1/60-1/2+1/19]
=1/2[5/54+1/20-1/51+2/57]
=1/2[201/2754+97/1140]
=1/2[496278/3139560]
=248139/3139560
=27571/348840