已知数列{an}满足a1=3,an+1-3an=3^n(n∈N^*),数列{bn}满足bn=an/3n,(1)证明数列{bn}是等比数列并求数列{bn}的通项公式,(2)求数列{an}的前n项和sn
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![已知数列{an}满足a1=3,an+1-3an=3^n(n∈N^*),数列{bn}满足bn=an/3n,(1)证明数列{bn}是等比数列并求数列{bn}的通项公式,(2)求数列{an}的前n项和sn](/uploads/image/z/1773283-67-3.jpg?t=%E5%B7%B2%E7%9F%A5%E6%95%B0%E5%88%97%EF%BD%9Ban%EF%BD%9D%E6%BB%A1%E8%B6%B3a1%3D3%2Can%2B1-3an%3D3%5En%28n%E2%88%88N%5E%2A%29%2C%E6%95%B0%E5%88%97%EF%BD%9Bbn%EF%BD%9D%E6%BB%A1%E8%B6%B3bn%3Dan%2F3n%2C%281%29%E8%AF%81%E6%98%8E%E6%95%B0%E5%88%97%EF%BD%9Bbn%EF%BD%9D%E6%98%AF%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97%E5%B9%B6%E6%B1%82%E6%95%B0%E5%88%97%EF%BD%9Bbn%EF%BD%9D%E7%9A%84%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F%2C%282%29%E6%B1%82%E6%95%B0%E5%88%97%EF%BD%9Ban%EF%BD%9D%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8Csn)
已知数列{an}满足a1=3,an+1-3an=3^n(n∈N^*),数列{bn}满足bn=an/3n,(1)证明数列{bn}是等比数列并求数列{bn}的通项公式,(2)求数列{an}的前n项和sn
已知数列{an}满足a1=3,an+1-3an=3^n(n∈N^*),数列{bn}满足bn=an/3n,(1)证明数列{bn}是等比数列并求数列{bn}的通项公式,(2)求数列{an}的前n项和sn
已知数列{an}满足a1=3,an+1-3an=3^n(n∈N^*),数列{bn}满足bn=an/3n,(1)证明数列{bn}是等比数列并求数列{bn}的通项公式,(2)求数列{an}的前n项和sn
(1)
a(n+1)-3an=3^n
a(n+1)/3^(n+1)-an/3^n=1/3
{an/3^n}是等差数列,d=1/3
bn-b1= (n-1)/3
bn = (n+2)/3
(2)
an = [(n+2)/3].3^n
= n(1/3)^(n-1) + 2.(1/3)^(n-1)
Sn=a1+a2+...+an
=[∑(i:1->n)i (1/3)^i ] + 3[1-(1/3)^n]
let
S = 1.(1/3)^0+2.(1/3)^1+...+n.(1/3)^(n-1) (1)
(1/3)S = 1.(1/3)^1+2.(1/3)^2+...+n.(1/3)^n (2)
(1)-(2)
(2/3)S = [1+ 1/3+...+1/3^(n-1)] - n(1/3)^n
= (3/2)(1-(1/3)^n) - n(1/3)^n
S =(9/4)(1-(1/3)^n) - (3/2).3^n
Sn = S +3[1-(1/3)^n]
=(9/4)(1-(1/3)^n) - (3/2).3^n + 3[1-(1/3)^n]