已知公差不为0的等差数列{an}满足a2=3,a1,a3,a7成等比数列 (1)求an的通项公式已知公差不为0的等差数列{an}满足a2=3,a1,a3,a7成等比数列(1)求an的通项公式数列{bn}满足bn=an/an+1+an+1/an,求数列bn前n
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![已知公差不为0的等差数列{an}满足a2=3,a1,a3,a7成等比数列 (1)求an的通项公式已知公差不为0的等差数列{an}满足a2=3,a1,a3,a7成等比数列(1)求an的通项公式数列{bn}满足bn=an/an+1+an+1/an,求数列bn前n](/uploads/image/z/7133463-63-3.jpg?t=%E5%B7%B2%E7%9F%A5%E5%85%AC%E5%B7%AE%E4%B8%8D%E4%B8%BA0%E7%9A%84%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97%EF%BD%9Ban%EF%BD%9D%E6%BB%A1%E8%B6%B3a2%3D3%2Ca1%2Ca3%2Ca7%E6%88%90%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97+%281%29%E6%B1%82an%E7%9A%84%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F%E5%B7%B2%E7%9F%A5%E5%85%AC%E5%B7%AE%E4%B8%8D%E4%B8%BA0%E7%9A%84%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97%EF%BD%9Ban%EF%BD%9D%E6%BB%A1%E8%B6%B3a2%3D3%2Ca1%2Ca3%2Ca7%E6%88%90%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97%281%29%E6%B1%82an%E7%9A%84%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F%E6%95%B0%E5%88%97%7Bbn%7D%E6%BB%A1%E8%B6%B3bn%3Dan%2Fan%2B1%2Ban%2B1%2Fan%2C%E6%B1%82%E6%95%B0%E5%88%97bn%E5%89%8Dn)
已知公差不为0的等差数列{an}满足a2=3,a1,a3,a7成等比数列 (1)求an的通项公式已知公差不为0的等差数列{an}满足a2=3,a1,a3,a7成等比数列(1)求an的通项公式数列{bn}满足bn=an/an+1+an+1/an,求数列bn前n
已知公差不为0的等差数列{an}满足a2=3,a1,a3,a7成等比数列 (1)求an的通项公式
已知公差不为0的等差数列{an}满足a2=3,a1,a3,a7成等比数列
(1)求an的通项公式
数列{bn}满足bn=an/an+1+an+1/an,求数列bn前n项的和Sn
设cn=2∧n(an+1/n-λ)若数列cn是单调递减数列求λ的取值范围
已知公差不为0的等差数列{an}满足a2=3,a1,a3,a7成等比数列 (1)求an的通项公式已知公差不为0的等差数列{an}满足a2=3,a1,a3,a7成等比数列(1)求an的通项公式数列{bn}满足bn=an/an+1+an+1/an,求数列bn前n
(Ⅰ)由题知 a\x0523 =a1a7,设等差数列{an}的公差为d,
则(a1+2d)2=a1(a1+6d),
a1d=2d2,∵d≠0
∴a1=2d. …(1分)
又∵a2=3,
∴a1+d=3a1=2,d=1…(2分)
∴an=n+1. …(3分)
(Ⅱ)∵bn= an\x05an+1 + an+1\x05an = n+1\x05n+2 + n+2\x05n+1 =2+ 1\x05n+1 - 1\x05n+2 . …(4分)
∴Sn=b1+b2+…+bn=(2+ 1\x052 - 1\x053 )+(2+ 1\x053 - 1\x054 )+…+(2+ 1\x05n+1 - 1\x05n+2 )=2n+ n\x052(n+2) . …(6分)
( III)cn=2n( an+1\x05n -λ)=2n( n+2\x05n -λ),使数列{cn}是单调递减数列,
则cn+1-cn=2n( 2(n+3)\x05n+1 - n+2\x05n -λ)<0对n∈N*都成立 …(7分)
即 2(n+3)\x05n+1 - n+2\x05n -λ<0⇒λ>( 2(n+3)\x05n+1 - n+2\x05n )max…(8分)
设f(n)= 2(n+3)\x05n+1 - n+2\x05n ,
f(n+1)-f(n)= 2(n+4)\x05n+2 - n+3\x05n+1 - 2(n+3)\x05n+1 + n+2\x05n
= 2(n+4)\x05n+2 + n+2\x05n - 3(n+3)\x05n+1
=2+ 4\x05n+2 +1+ 2\x05n -3- 6\x05n+1
= 2(2-n)\x05n(n+1)(n+2) …(9分)
∴f(1)<f(2)=f(3)>f(4)>f(5)>…
当n=2或n=3时,f(n)max= 4\x053 ,
∴( 2(n+3)\x05n+1 - n+2\x05n )max= 4\x053
所以λ> 4\x053 .