Question

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

Answer 1

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