1*2*3+2*3*4+...+n(n+1)(n+2)=在数列{an}中,an=1/(n+1)+2/(n+1)+...+n/(n+1).又bn=2/(an*an-1) 求数列{bn}的n项和sn=1/2+1/6+...+1/n(n+1) 若sn*sn+1=3/4
来源:学生作业帮助网 编辑:作业帮 时间:2024/06/17 00:50:49
![1*2*3+2*3*4+...+n(n+1)(n+2)=在数列{an}中,an=1/(n+1)+2/(n+1)+...+n/(n+1).又bn=2/(an*an-1) 求数列{bn}的n项和sn=1/2+1/6+...+1/n(n+1) 若sn*sn+1=3/4](/uploads/image/z/693610-34-0.jpg?t=1%2A2%2A3%2B2%2A3%2A4%2B...%2Bn%28n%2B1%29%28n%2B2%29%3D%E5%9C%A8%E6%95%B0%E5%88%97%7Ban%7D%E4%B8%AD%2Can%3D1%2F%28n%2B1%29%2B2%2F%28n%2B1%29%2B...%2Bn%2F%28n%2B1%29.%E5%8F%88bn%3D2%2F%28an%2Aan-1%29+%E6%B1%82%E6%95%B0%E5%88%97%7Bbn%7D%E7%9A%84n%E9%A1%B9%E5%92%8Csn%3D1%2F2%2B1%2F6%2B...%2B1%2Fn%28n%2B1%29+%E8%8B%A5sn%2Asn%2B1%3D3%2F4)
1*2*3+2*3*4+...+n(n+1)(n+2)=在数列{an}中,an=1/(n+1)+2/(n+1)+...+n/(n+1).又bn=2/(an*an-1) 求数列{bn}的n项和sn=1/2+1/6+...+1/n(n+1) 若sn*sn+1=3/4
1*2*3+2*3*4+...+n(n+1)(n+2)=
在数列{an}中,an=1/(n+1)+2/(n+1)+...+n/(n+1).又bn=2/(an*an-1) 求数列{bn}的n项和
sn=1/2+1/6+...+1/n(n+1) 若sn*sn+1=3/4
1*2*3+2*3*4+...+n(n+1)(n+2)=在数列{an}中,an=1/(n+1)+2/(n+1)+...+n/(n+1).又bn=2/(an*an-1) 求数列{bn}的n项和sn=1/2+1/6+...+1/n(n+1) 若sn*sn+1=3/4
n(n+1)(n+2)=n^3+3n^2+2n
1^3+2^3+...+n^3=[n(n+1)/2]^2(自然数立方和公式,可以用更高次的项来推,这里省略);
n项自然数平方和公式:n(n+1)(2n+1)/6;
则1*2*3+2*3*4+...+n(n+1)(n+2)=[n(n+1)/2]^2+n(n+1)(2n+1)/6+n(n+1)
这里还可以提取n(n+1),留给你了
2
先求出an=1/(n+1)+2/(n+1)+...+n/(n+1)=n/2
然后裂项相消
n>=2时
bn=2/(an*an-1)=8/[n(n-1)]=8[1/(n-1)-1/n]
则前n项和为8(n-1)/n;
3
1/n(n+1) =1/n-1/(n+1)
则sn=1-1/2+1/2-1/3+...+1/n-1/(n+1)=n/(n+1)
又sn*sn+1=n/(n+1)*(n+1)/(n+2)=n/(n+2)=3/4
解得n=6