1/3+1/5+1/7+1/9+1/11+1/33+1/35+1/45+1/55+1/77=?

1个回答

  • 1/3+1/5+1/7+1/9+1/11+1/33+1/35+1/45+1/55+1/77

    =1/3+1/5+1/7+1/9+1/11+(1/3-1/11)/8+(1/5-1/7)/2

    +(1/5-1/9)/4+(1/5-1/11)/6+(1/7-1/11)/4

    =1/3+1/5+1/7+1/9+1/11+(1/3)/8-(1/11)/8+(1/5)/2-(1/7)/2+

    (1/5)/4-(1/9)/4+(1/5)/6-(1/11)/6+(1/7)/4-(1/11)/4

    =(1/3)*[1+1/8]

    +(1/5)*[1+1/2+1/4+1/6]

    +(1/7)*[1-1/2+1/4]

    +(1/11)*[1-1/8-1/6-1/4]

    +(1/9)*[1-1/4]

    =(1/3)*(9/8)+(1/5)*(23/12)+(1/7)*(3/4)+(1/11)*(11/24)+(1/9)*(3/4)

    =(1/3)*(9/8)+(1/5)*(23/12)+(1/7)*(3/4)+1/24+1/12

    =(3/8)+(1/5)*(23/12)+(1/7)*(3/4)+1/8

    =1/2+(1/5)*(23/12)+(1/7)*(3/4)

    =(6*5*7+23*7+3*3*5)/(12*5*7)

    =[210+161+45]/(12*5*7)

    =416/(12*5*7)

    =104/105