Question

三道分数简便运算，急！！！！

①1/2×4+1/4×6+1/6×8+······+1/16×18

②1/3×6+1/6×9+······+1/18×21

③1/3+1/15+1/35+1/63+1/99

Answer 1

1.原式＝1/2(1/1Ⅹ1/2 1/2Ⅹ1/3……＋1/8Ⅹ1/9)＝1/2(1/1－1/2 1/2－1/3…… 1/8－1/9)=4/9 2原理同上 提三分之一出来后再如上变形
3 原式＝(1/(1×3) 1/(3×5) 1/(5×7) 1/(7Ⅹ9) 1/(9×11))
=(1/1－1/3 1/3－1/5 1/5－1/7 1/7－1/9 1/9－1/11)÷2
＝5/11

Answer 2

①公式1/n(n+2)=[1/n-1/(n+2)]/2
所以原式=(1/2-1/4+1/4-1/6+……+1/16-1/18)/2
=(1/2-1/18)/2
=2/9
②公式1/n(n+3)=[1/n-1/(n+3)]/3
所以原式=(1/3-1/6+1/6-1/9+……+1/18-1/21)/3
=(1/3-1/21)/3
=2/21
③1/3=1/1×3,1/15=1/3×5,……
所以原式=1/1×3+1/3×5+……+1/9×11
=(1-1/3+1/3-1/5+……+1/9-1/11)/2
=(1-1/11)/2
=5/11

Answer 3

①1/2×4+1/4×6+1/6×8+······+1/16×18
＝1/2×（1/2-1/4）+1/2×（1/4-1/6）+1/2×（1/6-1/8）+......+1/2×(1/16-1/18)
＝1/2×（1/2-1/4+1/4-1/6+1/6-1/8+......+1/16-1/18)
＝1/2×(1/2-1/18)
＝1/2×4/9
＝2/9

②1/3×6+1/6×9+······+1/18×21
＝1/3×(1/3-1/6)+1/3×(1/6-1/9)+.......+1/2×(1/18-1/21)
＝1/3×(1/3-1/6+1/6-1/9+.......+1/18-1/21)
＝1/3×(1/3-1/21)
＝1/3×3/7
＝1/7

③1/3+1/15+1/35+1/63+1/99
＝1/1×3+1/3×5+1/5×7+1/7×9+1/9×11
＝1/2×(1-1/3)+1/2×(1/3-1/5)+1/2×(1/5-1/7)+1/2×(1/7-1/9)+1/2×(1/9-1/11)
＝1/2×(1-1/3+1/3-1/5+1/5-1/7+1/7-1/9+1/9-1/11)
＝1/2×(1-1/11)
＝1/2×10/11
＝5/11

Answer 4

1.=1/2[(1/2-1/4)+(1/4-1/6)+...+(1/16-1/18)]
=(1/2)(1/2-1/18)
=（1/2）*（8/18）
=2/9

2.=1/3（1/3-1/6+1/6-1/9+...1/18-1/21）
=(1/3)*(1/3-1/21)
=(1/3)*(6/21)
=2/21

3.=1/1*3+1/3*5+1/5*7+1/7*9+1/9*11
=(1/2)(1-1/3+1/3-1/5+1/5-1/7+1/7-1/9+1/9-1/11)
=(1/2)(1-1/11)
=(1/2)(10/11)
=5/11