数学数列19已知数列{an}满足a1=1,an=3^(n-1)+a(n-1) (n≥2)1)求a2,a32)求an
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数学数列19
已知数列{an}满足a1=1,an=3^(n-1)+a(n-1) (n≥2)
1)求a2,a3
2)求an
已知数列{an}满足a1=1,an=3^(n-1)+a(n-1) (n≥2)
1)求a2,a3
2)求an
![数学数列19已知数列{an}满足a1=1,an=3^(n-1)+a(n-1) (n≥2)1)求a2,a32)求an](/uploads/image/z/16215270-6-0.jpg?t=%E6%95%B0%E5%AD%A6%E6%95%B0%E5%88%9719%E5%B7%B2%E7%9F%A5%E6%95%B0%E5%88%97%EF%BD%9Ban%EF%BD%9D%E6%BB%A1%E8%B6%B3a1%3D1%2Can%3D3%5E%28n-1%29%2Ba%28n-1%29+%EF%BC%88n%E2%89%A52%EF%BC%891%EF%BC%89%E6%B1%82a2%2Ca32%EF%BC%89%E6%B1%82an)
an=3^(n-1)+a(n-1) (两边同除以3^n)
an/3^n=a(n-1)/3*3^(n-1)+1/3
an/3^n-1/2=(1/3)[a(n-1)/3^(n-1)-1/2]
(an/3^n-1/2)/[a(n-1)/3^(n-1)-1/2]=1/3
所以{an/3^n-1/2}是以a1/3-1/2=-1/6为首相q=1/3为公比的等比数列
an/3^n-1/2=(-1/6)*(1/3)^(n-1)=-(1/3)^n/2=-3^(-n)/2
an=-1/2+3^n/2
a2=-1/2+9/2=4
a3=-1/2+27/2=13
an/3^n=a(n-1)/3*3^(n-1)+1/3
an/3^n-1/2=(1/3)[a(n-1)/3^(n-1)-1/2]
(an/3^n-1/2)/[a(n-1)/3^(n-1)-1/2]=1/3
所以{an/3^n-1/2}是以a1/3-1/2=-1/6为首相q=1/3为公比的等比数列
an/3^n-1/2=(-1/6)*(1/3)^(n-1)=-(1/3)^n/2=-3^(-n)/2
an=-1/2+3^n/2
a2=-1/2+9/2=4
a3=-1/2+27/2=13
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