1/2+1/6+1/12+1/20+1/30+1/42的求和公式是怎么推导的？为什么是N/N+1？

1/2+1/6+1/12+1/20+1/30+1/42 =1/(1*2)+1/(2*3)+1/(3*4)+1/(4*5)+1/(5*6)+1/(6*7) =(1/1-1/2)+(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+(1/5-1/6)+(1/6-1/7) =1-1/7 =6/