设数列{an}满足a1+3a2+3的平方a3+.+3的n-1次方an=n/3. (1)求数列{an}的通项.(2)设bn=n/an,求数列{bn}的前n项和sn
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![设数列{an}满足a1+3a2+3的平方a3+.+3的n-1次方an=n/3. (1)求数列{an}的通项.(2)设bn=n/an,求数列{bn}的前n项和sn](/uploads/image/z/1240624-64-4.jpg?t=%E8%AE%BE%E6%95%B0%E5%88%97%7Ban%7D%E6%BB%A1%E8%B6%B3a1%2B3a2%2B3%E7%9A%84%E5%B9%B3%E6%96%B9a3%2B.%2B3%E7%9A%84n-1%E6%AC%A1%E6%96%B9an%3Dn%2F3.+%EF%BC%881%EF%BC%89%E6%B1%82%E6%95%B0%E5%88%97%7Ban%7D%E7%9A%84%E9%80%9A%E9%A1%B9.%EF%BC%882%EF%BC%89%E8%AE%BEbn%3Dn%2Fan%2C%E6%B1%82%E6%95%B0%E5%88%97%7Bbn%7D%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8Csn)
设数列{an}满足a1+3a2+3的平方a3+.+3的n-1次方an=n/3. (1)求数列{an}的通项.(2)设bn=n/an,求数列{bn}的前n项和sn
设数列{an}满足a1+3a2+3的平方a3+.+3的n-1次方an=n/3. (1)求数列{an}的通项.
(2)设bn=n/an,求数列{bn}的前n项和sn
设数列{an}满足a1+3a2+3的平方a3+.+3的n-1次方an=n/3. (1)求数列{an}的通项.(2)设bn=n/an,求数列{bn}的前n项和sn
1) a1+3a2+3^2a3+3^3a4+.+3^(n-2)a(n-1)+3^(n-1)an=n/3
a1+3a2+3^2a3+3^3a4+.+3^(n-2)a(n-1)=(n-1)/3
两式相减,得
3^(n-1)an=n/3-(n-1)/3=1/3
an=(1/3)/3^(n-1)=1/(3^n)
2) bn=n/an=n/[1/(3^n)]=n*3^n
Sn=b1+b2+b3+.+bn
=1*3^1+2*3^2+3*3^3+.+n*3^n ------- ①
两边同乘以3,得
3Sn=1*3^2+2*3^3+3*3^4+.+n*3^(n+1) -------- ②
① - ②【错位相减】,得
(1-3)Sn=1*3^1+(2*3^2-1*3^2)+(3*3^3-2*3^3)+.+[n*3^n-(n-1)*3^n]-n*3^(n+1)
=3^1+3^2+3^3+3^4+.+3^n - n*3^(n+1)
=3*(1-3^n)/(1-3) -n*3^(n+1)
-2Sn= -3/2*(1-3^n) -n*3^(n+1)
4Sn=3*(1-3^n)+2n*3^(n+1)
=3-3^(n+1)+2n*3^(n+1)
=3(2n3^n-3^n-1)
=3[(2n-1)3^n-1]
Sn=3/4*[(2n-1)3^n-1]