数列{An}中,A1=1,An>0,并且((n+1)*A(n+1))^2-n*An^2+An+1*An=0,求通项

Question

数列{An}中,A1=1,An>0,并且((n+1)*A(n+1))^2-n*An^2+An+1*An=0,求通项

Answer 1

((n+1)*A(n+1))^2-n*An^2+An+1*An=0
将上式因式分解的
[(n+1)*A(n+1)-n*A(n)][(n+1)*A(n+1)+ An]=0
所以 (n+1)*A(n+1)-n*A(n)=0 或(n+1)*A(n+1)+ An=0
故 A(n+1)/An=n/(n+1)或A(n+1)/An=-1/(n+1)
∵An>0
∴A(n+1)/An=n/(n+1) 另一种不符要求,舍去
即 An/A(n-1)=(n-1)/n
An=[An/A(n-1)]*[A(n-1)/A(n-2)]*……*[A2/A1]*A1
=(n-1)/n * (n-2)/(n-1)*……*1/2 *A1
=1/n *1
=1/n
故所求通项公式为An=1/n
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