an满足a1=1 a2=2/3 且2an-1an+1=an(an-1+an+1) 求an的通向公式已知f(x)=2x/(x^2+1) x1=1/2 xn+1=f(x) 求证(x1-x2)^2/x1x2 +(x2-x3)^2/x2x3 +……+(xn-xn+1)^2/xnxn+1
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an满足a1=1 a2=2/3 且2an-1an+1=an(an-1+an+1) 求an的通向公式
已知f(x)=2x/(x^2+1) x1=1/2 xn+1=f(x) 求证
(x1-x2)^2/x1x2 +(x2-x3)^2/x2x3 +……+(xn-xn+1)^2/xnxn+1
1.
2an-1an+1=an(an-1+an+1)
2= an/an+1 + an/an-1
两边同乘1/an
2/an = 1/an+1 + 1 /an-1
等差中项原理
所以
{1/an}是 首项为 1 公差为 1/2 的等差数列
所以 1/ an = 1+ (n-1)*1/2 = 1/2 - (1/2)*n
an= 2/(1+n)
第一题用数学归纳法,或者化成等差数列之类的线性递推数列
an=2/(n+1)
第二题用放缩法,转化成可以裂项求和的线性递推数列即可
2a(n-1)a(n+1)=an[a(n-1)+a(n+1)]
2/an=1/a(n-1)+1/a(n+1)
设bn=1/an则an=1/bn,b1=1,b2=3/2,
2bn=b(n-1)+b(n+1) ,b2-b1=1/2
bn为首项为1,公差为1/2的等差数列,
bn=1+(1/2)(n-1)=(n+1)/2
an=1/bn=2/(n+1).
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