数列an满足a1=1,an+1=2(n+1)方*an/an+2n方,数列2n方/an为等差数列,求数列an的通项公式.
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数列an满足a1=1,an+1=2(n+1)方*an/an+2n方,数列2n方/an为等差数列,求数列an的通项公式.
数列an满足a1=1,an+1=2(n+1)方*an/an+2n方,数列2n方/an为等差数列,求数列an的通项公式.
数列an满足a1=1,an+1=2(n+1)方*an/an+2n方,数列2n方/an为等差数列,求数列an的通项公式.
an+1=2(n+1)^2*an/(an+2n^2)
右边分子分母同时除以an
an+1=2(n+1)^2/(1+2n^2/an)
然后将右面的分母乘到左面,左面的式子除到右面
1+2n^2/an=2(n+1)^2/(an+1)
已知2n^2/an是等差数列,由上式可以知道公差为1,
又知道a1=1,
所以:2/1=2,首项是2
2n^2/an的通项为n+1
an=2n^2/(n+1)
a(n+1) =2(n+1)^2.an/(an+2n^2)
1/a(n+1) = (an+2n^2)/[2(n+1)^2.an]
(n+1)^2/a(n+1)=(an+2n^2)/(2an)
= 1/2 + n^2/an
(n+1)^2/a(n+1)-n^2/an =1/2
n^2/an - 1/an = (n-1)/2
n^2/an = (n+1)/2
an = 2n^2/(n+1)
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