Question

定义数列an,an=3/2,an={a(n-1)+n-1,n为奇数/3a(n-1),n为偶数 （1）记bn=a(2n-1

Answer 1

下标用"[ ]"

1）因为(2n-1)是奇数,a[2n-1] =a[2n-1-1]+ 2n-1-1= a[2n-2]+2n-2

因为(2n-2)是偶数,a[2n-2]=3a[2n-2-1]=3a[2n-3]

那么,a[2n-1]=3a[2n-3]+2n-2

b[n]=a[2n-1]+n+1/2 =3a[2n-3]+2n-2+n+1/2=3*{a[2n-3]+n-1 + 1/2} =3*b[n-1]

{b[n]}是等比数列,b[1]=a[1]+1+1/2=3,公比为3

2）b[n]=3^n,即a[2n-1]+n+1/2=3^n

设N=2n-1为奇数,a[N]= 3^((N+1)/2) - N/2 -1

又因为N为奇数时有,a[N]=3a[N-1]

所以当N为偶数时有,a[N]=a[N+1]/3=3^(N/2) - N/6 -1/2

S[2n]= { a[1]+a[3]+...+a[2n-1] } +{ a[2]+a[4]+...+a[2n] }=3^(n+1)-3-(2n^2+5n)/3

(S[2n]+3)/3^n =3- (2n^2+5n)/3^(n+1)

(S[2(n+1)]+3)/3^(n+1)是将上式的n换成n+1,

通过化简,得到

当n>=1时(S[2(n+1)]+3)/3^(n+1) > (S[2n]+3)/3^n