1/1*3+1/2*4+1/3*5+1/4*6+1/5*7+…+1/18*20+1/38

通式1/1*3+1/2*4+1/3*5+....+1/n(n+2)
=1/2(1/1-1/3)+1/2(1/2-1/4)+1/2(1/3-1/5)+1/2[1/n-1/(n+2)]
=1/2[1-1/3+1/2-1/4+1/3-1/5+....+1/n-1/(n+2)]
=1/2[1+1/2-1/n-1/(n+2)]
所以 原式=1/2[1+1/2-1/18-1/20]+1/38
=251/180+1/38
最后结果很难算!!!!!!!!!!!!!!!!!!!

1/(2n-1)(2n+1)=1/2(1/2n-1 -1/2n+1)
1/2(1-1/20)+1/38=19/40+1/38=381/760

可以这么解题：
1/1*3+1/2*4+1/3*5+1/4*6+1/5*7+…+1/18*20+1/38
=1/2 *（1-1/3 + 1/2-1/4 + 1/3+1/5 + 1/4-1/6.+1/17-1/19 + 1/18-1/20) + 1/38
=1/2 * (1+1/2-1/19-1/20) + 1/38
=3/4-39/760+1/38
=3/4-19/760