设数列{an},{bn}满足;a1=4 a2=5/2,an+1=an+bn/2,bn+1=2anbn/an+bn 用数列
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设数列{an},{bn}满足;a1=4 a2=5/2,an+1=an+bn/2,bn+1=2anbn/an+bn 用数列an表示an+1;并证明;任意n属于
设数列{an},{bn}满足;a1=4 a2=5/2,an+1=an+bn/2,bn+1=2anbn/an+bn
(1)用数列an表示an+1;并证明;任意n属于N*都有an>2
(2)证明{lnan+2/an-2}是等比数列
a[n+1]=(an+bn)/2,b[n+1]=2anbn(/an+bn 是这样的 {ln[an+2/an-2]}
{ln[an+2/an-2]},这个是 ln((a[n]+2)/(a[n]-2))
设数列{an},{bn}满足;a1=4 a2=5/2,an+1=an+bn/2,bn+1=2anbn/an+bn
(1)用数列an表示an+1;并证明;任意n属于N*都有an>2
(2)证明{lnan+2/an-2}是等比数列
a[n+1]=(an+bn)/2,b[n+1]=2anbn(/an+bn 是这样的 {ln[an+2/an-2]}
{ln[an+2/an-2]},这个是 ln((a[n]+2)/(a[n]-2))
a[n+1]=(an+bn)/2
令n=1
bn=2a[n+1]-an
b1=2*5/2-4=1
(2)
a[n+1]=(an+bn)/2
b[n+1]=2anbn/(an+bn)
两式相乘有
a[n+1]b[n+1]=anbn=a1b1=4*1=4
同时,有:
b[n+1]=2anbn/(an+bn)
2/b[n+1]=1/an+1/bn=.
因anbn=4所以上式就是:
a[n+1]=4/b[n+1]=2/an+an/2
an为正,则a[n+1]肯定也为正
所以,有 a[n+1]=4/b[n+1]=2/an+an/2>=2
(2)
(a[n+1]+2)/(a[n+1]-2)
=(2a[n+1]+4)/(2a[n+1]-4)
=(an+bn+4)/(an+bn-4)
=(an^2+anbn+4an)/(an^2+anbn-4)
anbn=4
所以上式
=(an^2+4+4an)/(an^2+4-4an)
=((an+2)/(an-2))^2
即:
(a[n+1]+2)/(a[n+1]-2)=((an+2)/(an-2))^2
两边取对数有
ln((a[n+1]+2)/(a[n+1]-2))=2ln((an+2)/(an-2))
ln((a1+2)/(a1-2))=ln3
即
{ln((a[n]+2)/(a[n]-2))}是首项为ln3,公比为2的等比数列.
令n=1
bn=2a[n+1]-an
b1=2*5/2-4=1
(2)
a[n+1]=(an+bn)/2
b[n+1]=2anbn/(an+bn)
两式相乘有
a[n+1]b[n+1]=anbn=a1b1=4*1=4
同时,有:
b[n+1]=2anbn/(an+bn)
2/b[n+1]=1/an+1/bn=.
因anbn=4所以上式就是:
a[n+1]=4/b[n+1]=2/an+an/2
an为正,则a[n+1]肯定也为正
所以,有 a[n+1]=4/b[n+1]=2/an+an/2>=2
(2)
(a[n+1]+2)/(a[n+1]-2)
=(2a[n+1]+4)/(2a[n+1]-4)
=(an+bn+4)/(an+bn-4)
=(an^2+anbn+4an)/(an^2+anbn-4)
anbn=4
所以上式
=(an^2+4+4an)/(an^2+4-4an)
=((an+2)/(an-2))^2
即:
(a[n+1]+2)/(a[n+1]-2)=((an+2)/(an-2))^2
两边取对数有
ln((a[n+1]+2)/(a[n+1]-2))=2ln((an+2)/(an-2))
ln((a1+2)/(a1-2))=ln3
即
{ln((a[n]+2)/(a[n]-2))}是首项为ln3,公比为2的等比数列.
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