设Sn=1*2/1+2*3/1+3*4/1+.n*(n+1)/1,写出S1,S2,S3,S4的值,归纳并猜想出结果_数学解答

设Sn=1*2/1+2*3/1+3*4/1+.n*(n+1)/1,写出S1,S2,S3,S4的值,归纳并猜想出结果

人气：158 ℃　时间：2020-05-17 21:01:23

解答

1是分子吗?
S1=1/（182）=1/2,
S2=1/1*2+1/2*3=1-1/2+1/2-1/3=1-1/3=2/3,
S3=1/1*2+1/2*3+1/3*4
=1/1-1/2+1/2-1/3+1/3-1/4=1-1/4=3/4,
S4=1/1*2+1/2*3+1/3*4+1/4*5
=1/1-1/2+1/2-1/3+1/3-1/4+1/4-1/5
=1-1/5=4/5,
.
Sn=1/1*2+1/2*3+1/3*4+1/4*5+.+1/[n*(n+1)]
=1/1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+.+1/n-/(n+1)
=1-1/(n+1)
=n/(n+1).