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)+n+1/2,n属于正整数求证数列bn是等比数列;(2)记S2n=a1+a2+……+a(2n-1)+a2n,试比较(S2(n+1)+3)/3^(n+1)与(S2n+3)/3^n的大小,并说明
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![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)+n+1/2,n属于正整数求证数列bn是等比数列;(2)记S2n=a1+a2+……+a(2n-1)+a2n,试比较(S2(n+1)+3)/3^(n+1)与(S2n+3)/3^n的大小,并说明](/uploads/image/z/1349448-24-8.jpg?t=%E5%AE%9A%E4%B9%89%E6%95%B0%E5%88%97an%2Can%3D3%2F2%2Can%3D%7Ba%28n-1%29%2Bn-1%2Cn%E4%B8%BA%E5%A5%87%E6%95%B0%2F3a%28n-1%29%2Cn%E4%B8%BA%E5%81%B6%E6%95%B0+%EF%BC%881%EF%BC%89%E8%AE%B0bn%3Da%282n-1%29%2Bn%2B1%2F2%2Cn%E5%B1%9E%E4%BA%8E%E6%AD%A3%E6%95%B4%E6%95%B0%E6%B1%82%E8%AF%81%E6%95%B0%E5%88%97bn%E6%98%AF%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97%EF%BC%9B%EF%BC%882%EF%BC%89%E8%AE%B0S2n%3Da1%2Ba2%2B%E2%80%A6%E2%80%A6%2Ba%282n-1%29%2Ba2n%2C%E8%AF%95%E6%AF%94%E8%BE%83%EF%BC%88S2%EF%BC%88n%2B1%EF%BC%89%2B3%EF%BC%89%2F3%5E%28n%2B1%29%E4%B8%8E%28S2n%2B3%29%2F3%5En%E7%9A%84%E5%A4%A7%E5%B0%8F%2C%E5%B9%B6%E8%AF%B4%E6%98%8E)
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)+n+1/2,n属于正整数求证数列bn是等比数列;(2)记S2n=a1+a2+……+a(2n-1)+a2n,试比较(S2(n+1)+3)/3^(n+1)与(S2n+3)/3^n的大小,并说明
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)+n+1/2,n属于正整数
求证数列bn是等比数列;
(2)记S2n=a1+a2+……+a(2n-1)+a2n,试比较(S2(n+1)+3)/3^(n+1)与(S2n+3)/3^n的大小,并说明理由.
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)+n+1/2,n属于正整数求证数列bn是等比数列;(2)记S2n=a1+a2+……+a(2n-1)+a2n,试比较(S2(n+1)+3)/3^(n+1)与(S2n+3)/3^n的大小,并说明
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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