有穷数列{an}共有2k项(整数k≥2),a1=2,a(n+1)=(a-1)Sn+2(n=1,2,...,2k-1),其中常数a>1 求:若a=2^(2/(2k-1)),数列{bn}满足bn=(1/n)x log2(a1a2...an),(n=1,2,...,2k),求数列{bn}的通项公式
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![有穷数列{an}共有2k项(整数k≥2),a1=2,a(n+1)=(a-1)Sn+2(n=1,2,...,2k-1),其中常数a>1 求:若a=2^(2/(2k-1)),数列{bn}满足bn=(1/n)x log2(a1a2...an),(n=1,2,...,2k),求数列{bn}的通项公式](/uploads/image/z/8506530-18-0.jpg?t=%E6%9C%89%E7%A9%B7%E6%95%B0%E5%88%97%7Ban%7D%E5%85%B1%E6%9C%892k%E9%A1%B9%EF%BC%88%E6%95%B4%E6%95%B0k%E2%89%A52%EF%BC%89%2Ca1%3D2%2Ca%28n%2B1%29%3D%28a-1%29Sn%2B2%EF%BC%88n%3D1%2C2%2C...%2C2k-1%29%2C%E5%85%B6%E4%B8%AD%E5%B8%B8%E6%95%B0a%3E1+%E6%B1%82%EF%BC%9A%E8%8B%A5a%3D2%5E%282%EF%BC%8F%282k-1%29%29%2C%E6%95%B0%E5%88%97%7Bbn%7D%E6%BB%A1%E8%B6%B3bn%3D%EF%BC%881%EF%BC%8Fn%EF%BC%89x+log2%28a1a2...an%EF%BC%89%2C%EF%BC%88n%3D1%2C2%2C...%2C2k%29%2C%E6%B1%82%E6%95%B0%E5%88%97%7Bbn%7D%E7%9A%84%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F)
有穷数列{an}共有2k项(整数k≥2),a1=2,a(n+1)=(a-1)Sn+2(n=1,2,...,2k-1),其中常数a>1 求:若a=2^(2/(2k-1)),数列{bn}满足bn=(1/n)x log2(a1a2...an),(n=1,2,...,2k),求数列{bn}的通项公式
有穷数列{an}共有2k项(整数k≥2),a1=2,a(n+1)=(a-1)Sn+2(n=1,2,...,2k-1),其中常数a>1 求:
若a=2^(2/(2k-1)),数列{bn}满足bn=(1/n)x log2(a1a2...an),(n=1,2,...,2k),求数列{bn}的通项公式
有穷数列{an}共有2k项(整数k≥2),a1=2,a(n+1)=(a-1)Sn+2(n=1,2,...,2k-1),其中常数a>1 求:若a=2^(2/(2k-1)),数列{bn}满足bn=(1/n)x log2(a1a2...an),(n=1,2,...,2k),求数列{bn}的通项公式
当n=1时,a2=2a,a2/a1=a;
当2≤n≤2k-1时,an+1=(a-1)Sn+2,an=(a-1)Sn-1+2
∴an+1-an=(a-1)an
∴an+1/an=a
∴数列{an}是首项为2,公比为a的等比数列
∴an=2a^n-1 又a=2^[2/(2k-1)]
∴a1×a2×…an=2^na^[1+2+…+(n-1)]=2^n×a^[n(n-1) /2]=2^[n+n(n-1) /2k-1]
bn=1/n×[n+n(n-1) /2k-1=[n-1/2k-1]+1(n=1,2,...,2k)
bn=1/n[n+n(n-1) /(2k-1 )] ]=(n-1)/(2k-1)+1(n=1,2,……2k).