已知数列{an}中,a1=1,an=2n/(n-1) ·a +n, (n≥2,n∈正整数), n-1且bn=an/n+ん为等比数列.(1)求实数ん及{bn}的通项公式(2)求数
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![已知数列{an}中,a1=1,an=2n/(n-1) ·a +n, (n≥2,n∈正整数), n-1且bn=an/n+ん为等比数列.(1)求实数ん及{bn}的通项公式(2)求数](/uploads/image/z/5456577-57-7.jpg?t=%E5%B7%B2%E7%9F%A5%E6%95%B0%E5%88%97%EF%BD%9Ban%EF%BD%9D%E4%B8%AD%2Ca1%3D1%2Can%3D2n%2F%28n-1%29+++%C2%B7a+++++++++%2Bn%2C+++%EF%BC%88n%E2%89%A52%2Cn%E2%88%88%E6%AD%A3%E6%95%B4%E6%95%B0%EF%BC%89%2C++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++n-1%E4%B8%94bn%3Dan%2Fn%2B%E3%82%93%E4%B8%BA%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97.%EF%BC%881%EF%BC%89%E6%B1%82%E5%AE%9E%E6%95%B0%E3%82%93%E5%8F%8A%EF%BD%9Bbn%EF%BD%9D%E7%9A%84%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F%EF%BC%882%EF%BC%89%E6%B1%82%E6%95%B0)
已知数列{an}中,a1=1,an=2n/(n-1) ·a +n, (n≥2,n∈正整数), n-1且bn=an/n+ん为等比数列.(1)求实数ん及{bn}的通项公式(2)求数
已知数列{an}中,a1=1,an=2n/(n-1) ·a +n, (n≥2,n∈正整数),
n-1
且bn=an/n+ん为等比数列.
(1)求实数ん及{bn}的通项公式
(2)求数列{an}的前n项和.
真心速求 求解答过程,
已知数列{an}中,a1=1,an=2n/(n-1) ·a +n, (n≥2,n∈正整数), n-1且bn=an/n+ん为等比数列.(1)求实数ん及{bn}的通项公式(2)求数
(1)a(n+1)=(2n+2)/n*an+(n+1)
则b(n+1)=a(n+1)/(n+1)+ん=2an/n+1+ん
因为bn=an/n+ん为等比数列,故设b(n+1)=qbn
则2an/n+1+ん=q(an/n+ん)=qan/n+qん
系数对比知q=2 且1+ん=qん
则ん=1
故{bn}是首项为b1=a1+ん=2 公比为2
故bn=2^n
(2)由(1)得2^n=an/n+1 则an=n*2^n+n
用分组求和以及错位相减法即可求得数列{an}的前n项和(自己算吧)