3/1×3+3/3×5+3/5×7+...+3/19×21

题目

3/1×3+3/3×5+3/5×7+...+3/19×21

答案

1/n(n+2)=1/2[1/n-1/(n+2)]
3/1×3+3/3×5+3/5×7+...+3/19×21
=3(1/1*3+1/3*5+1/5*7+.+1/19*21)
=3/2(1-1/3+1/3-1/5+1/5-1/7+.+1/19-1/21)
=3/2(1-1/21)
=(3/2)*(20/21)
=10/7