Question

小学奥数题-分数简算 好难啊……

（7的平方-1）分之（7的平方+1）+（9的平方-1）分之（9的平方+1）+（11的平方-1）分之（11的平方+1）+……+（99的平方-1）分之（99的平方+1）=？
求详细过程~多谢！

Answer 1

=1+2/(7²-1)+1+2/(9²-1)+…+1+2/(99²-1)
=(99-7)÷2+1+[2/(6×8)+2/(8×10)+…+2/(98×100)]
=47+(1/6-1/8+1/8-1/10+…+1/98-1/100)
=47+(1/6-1/100)
=47+47/300
=47又47/300

Answer 2

=(7+1)^2/(7^2-1)-2*7*1/(7^2-1)+(9+1)^2/(9^2-1)-2*9*1/(9^2-1)+...+(99+1)^2/(99^2-1)-2*99*1/(99^2-1)
=8^2/[(7-1)(7+1)]+10^2/[(9-1)(9+1)]+...+100^2/[(99-1)(99+1)]-2{7/[(7-1)(7+1)]+9/[(9-1)9+1)]+...+99/[(99-1)(99+1)]}
=8/6+10/8+...+100/98-2[7(1/6-1/8)/2+9(1/8-1/10)/2+...+99(1/98-1/100)/2]
=8/6+10/8+...+100/98-[7(1/6-1/8)+9(1/8-1/10)+...99(1/98-1/100)]
=8/6+10/8+...+100/98-(7/6-7/8+9/8-9/10+...+99/98-99/100)
=8/6+10/8+...+100/98-(7/6+2/8+2/10+...+2/98-99/100)
=(8/6-7/6)+(10/8-2/8)+...+(100/98-2/98)+99/100
=1/6+1+1+...+1+99/100
=1/6+(100-8)/2+99/100
=1/6+46+99/100

Answer 3

(7*7+1)/(7*7-1)=1+2/(7*7-1)=1+2/(6*8)=1+2/(2*2*3*4)=1+0.5*1/(3*4)=1+0.5*(1/3-1/4)(9*9+1)/(9*9-1)=1+2/(9*9-1)=1+2/(8*10)=1+2/(2*2*4*5)=1+0.5*1/(4*5)=1+0.5*(1/4-1/5)
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(99*99+1)/(99*99-1)=1+0.5*(1/98-1/100)

从7到99一共有47个奇数，所以(7*7+1)/(7*7-1)+(9*9+1)/(9*9-1)+(11*11+1)/(11*11-1)+........(99*99+1)/(99*99-1)=47+0.5*(1/3-1/4+1/4-1/5+1/5-......+1/49-1/50)=47+0.5*(1/3-1/50)=47+47/300

Answer 4

结果是47+1/6-1/100