Question

数学问题（简便计算）

1×1/2+1/2×1/3+1/3×1/4+1/4×1/5+1/5×1/6+1/6×1/7

（1/4×1/7+1/7×1/10+1/10×1/13+1/13×1/16)×3

Answer 1

第一题=1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7=1-1/7=6/7
因为1*1/2=1-1/2 ，后面一样把乘号变减号就可以。
第二题其实差不多
1/4*1/7=1/3*(1/4-1/7)也就是说把乘号变减号后还要提一个1/3出来，那后面又乘以3，所以结果为：1/3*（1/4-1/7+1/7-1/10+1/10-1/13+1/13-1/16)*3=1/4-1/16=3/16

Answer 2

1×1/2+1/2×1/3+1/3×1/4+1/4×1/5+1/5×1/6+1/6×1/7

=（1-1/2）+（1/2-1/3）+（1/3-1/4）+……+(1/6-1/7)

=1-1/7

=6/7

（1/4×1/7+1/7×1/10+1/10×1/13+1/13×1/16)×3

=(1/4-1/7)+(1/7-1/10)+……+（1/13-1/16）

=1/4-1/16

=3/16

Answer 3

解：
(1）1×1/2+1/2×1/3+1/3×1/4+1/4×1/5+1/5×1/6+1/6×1/7
=1-1/2+1/2-1/3+1/3-1/4+……-1/7
=1-1/7
=6/7
(2)（1/4×1/7+1/7×1/10+1/10×1/13+1/13×1/16)×3
=1/4-1/7+1/7-1/10+……+1/13-1/16
=1/4-1/16
=3/16

Answer 4

记住公式 1/n - 1/(n+1) = 1/n * 1/(n+1)
1/n - 1/(n+k) = (1/n-1/(n+k))/k

根据公式展开得
原式: 1/1-1/2+1/2-1/3+1/3-1/4 ... +1/6-1/7 =1-1/7=6/7

第2题 2个数值均相差3 即k=3
代入公式后
原式： 1/3*（1/4-1/7+1/7-1/10+1/10-1/13+1/13-1/16）*3
=1/4-1/16
=3/16

Answer 5

因为1×1/2+1/2×1/3+1/3×1/4+1/4×1/5+1/5×1/6+1/6×1/7

=（1-1/2）+（1/2-1/3）+（1/3-1/4）+……+(1/6-1/7)

=1-1/7

=6/7

所以（1/4×1/7+1/7×1/10+1/10×1/13+1/13×1/16)×3

=(1/4-1/7)+(1/7-1/10)+……+（1/13-1/16）

=1/4-1/16

=3/16