设数列{an}满足关系an=3/2(an-1)+5(n≥2),a1=-17/2.令bn=an+10,求数列{bn}的前n项和Sn
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![设数列{an}满足关系an=3/2(an-1)+5(n≥2),a1=-17/2.令bn=an+10,求数列{bn}的前n项和Sn](/uploads/image/z/204588-36-8.jpg?t=%E8%AE%BE%E6%95%B0%E5%88%97%7Ban%7D%E6%BB%A1%E8%B6%B3%E5%85%B3%E7%B3%BBan%3D3%2F2%EF%BC%88an-1%EF%BC%89%2B5%EF%BC%88n%E2%89%A52%EF%BC%89%2Ca1%3D-17%2F2.%E4%BB%A4bn%3Dan%2B10%2C%E6%B1%82%E6%95%B0%E5%88%97%7Bbn%7D%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8CSn)
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设数列{an}满足关系an=3/2(an-1)+5(n≥2),a1=-17/2.令bn=an+10,求数列{bn}的前n项和Sn
设数列{an}满足关系an=3/2(an-1)+5(n≥2),a1=-17/2.令bn=an+10,求数列{bn}的前n项和Sn
设数列{an}满足关系an=3/2(an-1)+5(n≥2),a1=-17/2.令bn=an+10,求数列{bn}的前n项和Sn
先求出an的前n项和:an=3/2an-1+5,进行变换,等价于an+10=3/2(an-1+10),即可得
{an+10}是等比数列,公比1.5,首项为1.5
所以bn=1.5的n次方
sn=3-3*(1.5的n次方)
因为2an=3/2(an-1)+5(n≥2)两边都加10得到an+10=3/2*[(an-1)+10].....(1)
因为bn=an+10,所以bn-1(n-1是下标,下同)=an-1+10
由(1)得bn=3/2*bn-1(n≥2),所以数列bn是以b1=a1+10=3/2,公比是3/2的等比数列,Sn=b1(1-q^n)/(1-q)代入b1=q=3/2得到Sn=3^(n+1)2^n -3
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