1/3+1/8+1/15+1/24+...+1/360=

1个回答

  • 1/3=1/(1*3)=1/2*(1-1/3)

    1/8=1/(2*4)=1/2*(1/2-1/4)

    1/15=1/(3*5)=1/2*(1/3-1/5)

    1/24=1/(4*6)=1/2*(1/4-1/6)

    .

    1/323=1/(17*19)=1/2*(1/17-1/19)

    1/360=1/(18*20)=1/2*(1/18-1/20)

    1/3+1/8+1/15+1/24+...+1/360

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

    =1/2*(1+1/2-1/3+1/3-1/4+...-1/17+1/17-1/18+1/18-1/19-1/20)

    =1/2*(1+1/2-1/19-1/20)

    =1/2*531/380

    =531/760