设各项了均为正数的数列{an}的前n项和为Sn,已知a1=1,且(S(n+1)+λ)an=(Sn+1)a(n+1)对一切n∈N^*都成立.①若λ=1求数列an的通项公式②求λ的值.使数列an是等差数列
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![设各项了均为正数的数列{an}的前n项和为Sn,已知a1=1,且(S(n+1)+λ)an=(Sn+1)a(n+1)对一切n∈N^*都成立.①若λ=1求数列an的通项公式②求λ的值.使数列an是等差数列](/uploads/image/z/8817514-34-4.jpg?t=%E8%AE%BE%E5%90%84%E9%A1%B9%E4%BA%86%E5%9D%87%E4%B8%BA%E6%AD%A3%E6%95%B0%E7%9A%84%E6%95%B0%E5%88%97%EF%BD%9Ban%EF%BD%9D%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8C%E4%B8%BASn%2C%E5%B7%B2%E7%9F%A5a1%3D1%2C%E4%B8%94%EF%BC%88S%EF%BC%88n%2B1%EF%BC%89%2B%CE%BB%EF%BC%89an%3D%EF%BC%88Sn%2B1%EF%BC%89a%EF%BC%88n%2B1%EF%BC%89%E5%AF%B9%E4%B8%80%E5%88%87n%E2%88%88N%5E%2A%E9%83%BD%E6%88%90%E7%AB%8B.%E2%91%A0%E8%8B%A5%CE%BB%3D1%E6%B1%82%E6%95%B0%E5%88%97an%E7%9A%84%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F%E2%91%A1%E6%B1%82%CE%BB%E7%9A%84%E5%80%BC.%E4%BD%BF%E6%95%B0%E5%88%97an%E6%98%AF%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97)
设各项了均为正数的数列{an}的前n项和为Sn,已知a1=1,且(S(n+1)+λ)an=(Sn+1)a(n+1)对一切n∈N^*都成立.①若λ=1求数列an的通项公式②求λ的值.使数列an是等差数列
设各项了均为正数的数列{an}的前n项和为Sn,已知a1=1,且(S(n+1)+λ)an=(Sn+1)a(n+1)
对一切n∈N^*都成立.①若λ=1求数列an的通项公式②求λ的值.使数列an是等差数列
设各项了均为正数的数列{an}的前n项和为Sn,已知a1=1,且(S(n+1)+λ)an=(Sn+1)a(n+1)对一切n∈N^*都成立.①若λ=1求数列an的通项公式②求λ的值.使数列an是等差数列
(1)若λ=1,则(S(n+1)+λ)an=(Sn+1)a(n+1)两边除以ana(n+1)得
S(n+1)/a(n+1)+1/a(n+1)=Sn/an+1/an
∴Sn/an+1/an,是常数列.Sn/an+1/an=2
解得,Sn=2an-1,∴an=2^(n-1)
(2)a1=1,
n=1代入已知,得a2=1+λ
n=2代入已知,得a3=(1+λ)²
由an是等差数列.∴2a2=a1+a3,得λ=0
而当λ=0时,S(n+1)an=(Sn+1)a(n+1)两边除以ana(n+1)得,
S(n+1)/a(n+1)=Sn/an+1/an,
能满足an=1是等差数列.
所以λ=0.