已知数列｛an}中,a1=12a(n+1)-a(n)=2n则a(n)/n的最小值为?A4B2C3或4D6

Question

已知数列｛an}中,a1=12 a(n+1)-a(n)=2n则a(n)/n的最小值为?A4 B2 C3或4 D6

Answer 1

a(n+1)=a(n)+2n,
a(n+1)+n^2=a(n)+n^2+2n+1-1,
-1=[a(n+1)-(n+1)^2]-[a(n)-n^2],
{a(n)-n^2}是首项为a(1)-1^2=11,公差为-1的等差数列.
a(n)-n^2=11-(n-1)=12-n,
a(n)=12-n+n^2,
a(n)/n=n+12/n-1>=2(n*12/n)^(1/2)-1=2(12)^(1/2)-1,
等号成立当且仅当n=12/n,n=(12)^(1/2)=2*3^(1/2),3