等差数列{an}的各项均为正数,a1=3,前n项和为Sn,{bn}为等比数列,b1=1,且b2S2=64,b3S3=9601.求数列{an}与{bn}的通项公式(2)求1/S1+1/S2+...+1/Sn
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![等差数列{an}的各项均为正数,a1=3,前n项和为Sn,{bn}为等比数列,b1=1,且b2S2=64,b3S3=9601.求数列{an}与{bn}的通项公式(2)求1/S1+1/S2+...+1/Sn](/uploads/image/z/3623224-40-4.jpg?t=%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97%7Ban%7D%E7%9A%84%E5%90%84%E9%A1%B9%E5%9D%87%E4%B8%BA%E6%AD%A3%E6%95%B0%2Ca1%3D3%2C%E5%89%8Dn%E9%A1%B9%E5%92%8C%E4%B8%BASn%2C%7Bbn%7D%E4%B8%BA%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97%2Cb1%3D1%2C%E4%B8%94b2S2%3D64%2Cb3S3%3D9601.%E6%B1%82%E6%95%B0%E5%88%97%EF%BD%9Ban%EF%BD%9D%E4%B8%8E%EF%BD%9Bbn%EF%BD%9D%E7%9A%84%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F%EF%BC%882%EF%BC%89%E6%B1%821%2FS1%2B1%2FS2%2B...%2B1%2FSn)
等差数列{an}的各项均为正数,a1=3,前n项和为Sn,{bn}为等比数列,b1=1,且b2S2=64,b3S3=9601.求数列{an}与{bn}的通项公式(2)求1/S1+1/S2+...+1/Sn
等差数列{an}的各项均为正数,a1=3,前n项和为Sn,{bn}为等比数列,b1=1,且b2S2=64,b3S3=960
1.求数列{an}与{bn}的通项公式
(2)求1/S1+1/S2+...+1/Sn
等差数列{an}的各项均为正数,a1=3,前n项和为Sn,{bn}为等比数列,b1=1,且b2S2=64,b3S3=9601.求数列{an}与{bn}的通项公式(2)求1/S1+1/S2+...+1/Sn
1.
设等差数列{an}公差为d,等比数列{bn}公比为q,则:
b2S2=(b1*q)(a1+a1+d)=q(6+d)=64……①
b3S3=(b1*q^2)(a1+a1+d+a1+2d)=(q^2)(9+3d)=960即(q^2)(3+d)=320……②
联立①②得q=8或q=40/3,则d=2或d=-6/5(舍去),故d=2,q=8
∴an=a1+(n-1)d=2n+1
bn=b1*q^(n-1)=8^(n-1)=2^(3n-3)
2.
Sn=(a1+an)*n/2=n(n+2)
1/Sn=1/[n(n+2)]=(1/2)*[1/n-1/(n+2)]
∴原式=(1/2)*{(1-1/3)+(1/2-1/4)+(1/3-1/5)+…+[1/(n-2)-1/n]+[1/(n-1)-1/(n+1)]+[1/n-1/(n+2)]}
=(1/2)*[1+1/2-1/(n+1)-1/(n+2)]
=(1/2)*{3/2-(2n+3)/[(n+1)(n+2)]}
=3/4-(2n+3)/[2(n+1)(n+2)]
(1)
设an=a1+(n-1)d,d为公差,d≠0,
bn=b1q^(n-1),q≠0,1,则:
b2S2=b1q(a1+a2)=64
q(6+d)=64
b3S3=q^2 (a1+a2+a3)=q^2 (3a2)=960
联立:q=8,d=2
则:an=2n+1
bn=8^(n-1)
(2)
Sn=n(n...
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(1)
设an=a1+(n-1)d,d为公差,d≠0,
bn=b1q^(n-1),q≠0,1,则:
b2S2=b1q(a1+a2)=64
q(6+d)=64
b3S3=q^2 (a1+a2+a3)=q^2 (3a2)=960
联立:q=8,d=2
则:an=2n+1
bn=8^(n-1)
(2)
Sn=n(n+2)
1/Sn=1/2 * [1/n - 1/(n+2)]=
(1/S1)+(1/S2)+…(1/Sn)=1/2 *
[1/1-1/3+1/2-1/4+...+1/(n-1)-1/(n+1)+1/n-1/(n+2)]
=[1+1/2-1(n+1)-1/(n+2)]
=3/4 - 1/[2(n+1)] - 1/[2(n+2)]
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