已知数列{an}满足a1=4,a(n+1)=an+(k*3^n)+1(n∈N*,k为常数),a1,a2+6,a3成等差数列.(1)求k的值以及数列{an}的通项公式;(2)设数列{bn}满足bn=n/(an-n),求数列{bn}的前n项和Sn.(1)k=2;an=(3^n)+n(2)Sn=(3/
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![已知数列{an}满足a1=4,a(n+1)=an+(k*3^n)+1(n∈N*,k为常数),a1,a2+6,a3成等差数列.(1)求k的值以及数列{an}的通项公式;(2)设数列{bn}满足bn=n/(an-n),求数列{bn}的前n项和Sn.(1)k=2;an=(3^n)+n(2)Sn=(3/](/uploads/image/z/2782301-5-1.jpg?t=%E5%B7%B2%E7%9F%A5%E6%95%B0%E5%88%97%7Ban%7D%E6%BB%A1%E8%B6%B3a1%3D4%2Ca%28n%2B1%29%3Dan%2B%28k%2A3%5En%29%2B1%28n%E2%88%88N%2A%2Ck%E4%B8%BA%E5%B8%B8%E6%95%B0%29%2Ca1%2Ca2%2B6%2Ca3%E6%88%90%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97.%EF%BC%881%EF%BC%89%E6%B1%82k%E7%9A%84%E5%80%BC%E4%BB%A5%E5%8F%8A%E6%95%B0%E5%88%97%7Ban%7D%E7%9A%84%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F%EF%BC%9B%EF%BC%882%EF%BC%89%E8%AE%BE%E6%95%B0%E5%88%97%7Bbn%7D%E6%BB%A1%E8%B6%B3bn%3Dn%2F%EF%BC%88an-n%EF%BC%89%2C%E6%B1%82%E6%95%B0%E5%88%97%7Bbn%7D%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8CSn.%EF%BC%881%EF%BC%89k%3D2%EF%BC%9Ban%3D%283%5En%29%2Bn%EF%BC%882%EF%BC%89Sn%3D%283%2F)
已知数列{an}满足a1=4,a(n+1)=an+(k*3^n)+1(n∈N*,k为常数),a1,a2+6,a3成等差数列.(1)求k的值以及数列{an}的通项公式;(2)设数列{bn}满足bn=n/(an-n),求数列{bn}的前n项和Sn.(1)k=2;an=(3^n)+n(2)Sn=(3/
已知数列{an}满足a1=4,a(n+1)=an+(k*3^n)+1(n∈N*,k为常数),a1,a2+6,a3成等差数列.
(1)求k的值以及数列{an}的通项公式;
(2)设数列{bn}满足bn=n/(an-n),求数列{bn}的前n项和Sn.
(1)k=2;an=(3^n)+n
(2)Sn=(3/4)-[(2n+3)/4]*(1/3)^n
已知数列{an}满足a1=4,a(n+1)=an+(k*3^n)+1(n∈N*,k为常数),a1,a2+6,a3成等差数列.(1)求k的值以及数列{an}的通项公式;(2)设数列{bn}满足bn=n/(an-n),求数列{bn}的前n项和Sn.(1)k=2;an=(3^n)+n(2)Sn=(3/
A1=4, A1=A2-K*3-1, 责 A2=5+3K 同理 A3=6+12K 又:a1,a2+6,a3成等差数列.可得K=2
则:a1=a2+2*3^1-1
a2=a3+2*3^2-1
………………
A(n-1)=An+2*3^(n-1)
两边同时相加可得S(n-1)与Sn的关系 Sn-S(n-1)即为An
an+a
以n+1代n,得
a
相减得a
a1=1,a1+a2=1/4,a2=-3/4,
a<2k-1>=1+(1/4)(k-1)=(k+3)/4,
a<2k>=-3/4+(1/4)(k-1)=(k-4)/4.
总之,an=[-0.5+(-1)...
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an+a
以n+1代n,得
a
相减得a
a1=1,a1+a2=1/4,a2=-3/4,
a<2k-1>=1+(1/4)(k-1)=(k+3)/4,
a<2k>=-3/4+(1/4)(k-1)=(k-4)/4.
总之,an=[-0.5+(-1)^(n+1)*7.5+n]/8.
sn=a1+4a2+4^2a3+…+4^(n-1)an,①
16sn=..........4^2a1+...+4^(n-1)a
①-②,-15sn=1-3+(1/4)[4^2+4^3+……+4^(n-1)]-4^n[a
=-2-(1/12)(16-4^n)-4^n[-0.5+(-1)^n*7.5+n-1-2+(-1)^(n+1)*30+4n]/8
=(-1/12)(40-4^n)-4^n[5n-3.5-(-1)^n*22.5]/8
=(-1/24){80-4^n[15n-12.5-(-1)^n*67.5]},
∴5sn=(1/72){80-4^n[15n-12.5-(-1)^n*67.5]},
5sn-4^nan=(1/72){80-4^n[24n-17-(-1)^n*135 ]}.
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