一道【数列】解答题已知数列{an}满足an/an-1=(n+1)/(n-1),(n∈N*,n>1),a1=2注意:an-
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一道【数列】解答题
已知数列{an}满足an/an-1=(n+1)/(n-1),(n∈N*,n>1),a1=2
注意:an-1中的n-1是下标哦.
(1)求证:数列{an}的通项公式an=n(n+1)
(2)求数列{1/an}的前n项和Tn
已知数列{an}满足an/an-1=(n+1)/(n-1),(n∈N*,n>1),a1=2
注意:an-1中的n-1是下标哦.
(1)求证:数列{an}的通项公式an=n(n+1)
(2)求数列{1/an}的前n项和Tn
![一道【数列】解答题已知数列{an}满足an/an-1=(n+1)/(n-1),(n∈N*,n>1),a1=2注意:an-](/uploads/image/z/463379-59-9.jpg?t=%E4%B8%80%E9%81%93%E3%80%90%E6%95%B0%E5%88%97%E3%80%91%E8%A7%A3%E7%AD%94%E9%A2%98%E5%B7%B2%E7%9F%A5%E6%95%B0%E5%88%97%7Ban%7D%E6%BB%A1%E8%B6%B3an%2Fan-1%3D%EF%BC%88n%2B1%29%2F%28n-1%29%2C%EF%BC%88n%E2%88%88N%2A%2Cn%3E1%29%2Ca1%3D2%E6%B3%A8%E6%84%8F%EF%BC%9Aan-)
an/a(n-1)=(n+1)/(n-1)(n>=2)
a(n-1)/a(n-2)=n(n-2)
...
a2/a1=3/1
上式全部相乘
an/a1=(n+1)!/2(n-1)!=n(n+1)/2,
an=n(n+1)
1/an=1/n(n+1)=1/n-1/(n+1)
Tn=1/a1+1/a2+...+1/an
=1-1/2+1/2-1/3+...+1/n-1/(n+1)
=1-1/(n+1)
=n/(n+1)
a(n-1)/a(n-2)=n(n-2)
...
a2/a1=3/1
上式全部相乘
an/a1=(n+1)!/2(n-1)!=n(n+1)/2,
an=n(n+1)
1/an=1/n(n+1)=1/n-1/(n+1)
Tn=1/a1+1/a2+...+1/an
=1-1/2+1/2-1/3+...+1/n-1/(n+1)
=1-1/(n+1)
=n/(n+1)
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