Question

计算1/1*3+1/2*4+1/3*5+......+1/18*20

Answer 1

分数拆分公式：k/n(n+d)=k/d(1/n-1/(n+d))
1/1*3=1/2(1-1/3),1/2*4=1/2(1/2-1/4)
所以原式＝1/2(1－1/3+1/2-1/4+1/3-1/5+……+1/18-1/20)
＝1/2(1＋1/2-1/19-1/20)
＝531/760

Answer 2

1/1*3+1/2*4+1/3*5+......+1/18*20
=[1/1*3+1/3*5+...+1/17*19]+[1/2*4+1/4*6+...+1/18*20]
=[(1-1/3)/2+(1/3-1/5)/2+...+(1/17-1/19)/2]+[(1/2-1/4)/2+(1/4-1/6)/2+...+(1/18-1/20)/2]
=(1+1/2-1/19-1/20)/2
=531/760

Answer 3

1/(2n-1)(2n+1) + 1/2n(2n+2)

= { 1/(2n-1) - 1/ (2n+1) + 1/(2n) - 1/(2n+2) } /2

1/1*3+1/2*4+1/3*5+......+1/18*20

= { 1/1 -1/3 + 1/2 -1/4 + 1/3 - 1/5 + 1/4 -1/6 + 1/5 -1/7 + 1/6 - 1/8 + .... + 1/13 -1/15 + 1/14 - 1/16 + 1/15 - 1/17 + 1/16 -1/18 + 1/17 -1/19 + 1/18 -1/20 } /2

= { 1/1 -1/19 + 1/2 - 1/20} /2

= 531/760