原式=[15/3-(49/12-63/20+77/30-91/42+105/56)]÷43
=(-5/6+35/6-49/12+63/20-77/30+91/42-105/56)÷43
=[-5/6+7×(5/6-7/12+9/20-11/30+13/42-15/56)]÷43
=[-5/6+7×(1/2-1/4+1/4-1/6+1/6-1/8)]÷43
=(-5/6+21/8)÷43
=43/24÷43
=1/24
说明:
1.楼主给出的题目有误,最后应该是除以43;
2.(5/6-7/12+9/20-11/30+13/42-15/56)中的数是有规律的,每一项都可用An=(2n+1)/n(n+1),(n≧2)表示,例如:A1=(2×2+1)/2×3=5/6;
3.An-An+1=(2n+1)/n(n+1)-[2(n+1)+1]/(n+1)(n+2)
=[2(n+1)-1]/ n(n+1)- [2(n+2)-1]/(n+1)(n+2)
=2/n-1/n(n+1)-2/(n+1)+ 1/(n+1)(n+2)
=2/n-1/n+1/(n+1)-2/(n+1)+ 1/(n+1)- 1/ (n+2)
=1/n-1/ (n+2)
因此,5/6-7/12+9/20-11/30+13/42-15/56
=1/2-1/4+1/4-1/6+1/6-1/8
4.当题目()中的项比较多时,此方法更能显示其优越性.