数列满足a1=1/3,an+1=an+a2n/n2,证明:an

Question

数列满足a1=1/3,an+1=an+a2n/n2,证明:an

Answer 1

1.
a(n+1)=2(1+1／n)^2*an
a(n+1)=2[(n+1)／n]^2*an
a(n+1)/(n+1)^2=2*an／n^2
则an／n^2是首项为a1／1^2=a1=2,公比为2的等比数列.
an／n^2=2*2^(n-1)=2^n
an=(2^n)n^2.
2.
bn=an／n=n2^n
sn=1*2^1+2*2^2+3*2^3+……+(n-2)2^(n-2)+(n-1)2^(n-1)+n2^n
2sn=1*2^2+2*2^3+3*2^4+……+(n-2)2^(n-1)+(n-1)2^n+n2^(n+1)
两式相减：
-sn=2^1+2^2+2^3+……+2^(n-2)+2^(n-1)+2^n-n2^(n+1)
=2(2^n-1)/(2-1)-n2^(n+1)
=2^(n+1)-2-n2^(n+1)
=-2+(1-n)2^(n+1)
sn=2+(n-1)2^(n+1)