已知数列{an}中,a1=1,点(an,an+1)在函数y=3x+2的图像上(n属于N*),(1)证明:数列{an+1}是等比数列,(2)求数列{an}的前n项和

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已知数列{an}中,a1=1,点(an,an+1)在函数y=3x+2的图像上(n属于N*),(1)证明:数列{an+1}是等比数列,(2)求数列{an}的前n项和

(1)点(an,an+1)在函数y=3x+2的图像上(n属于N*)
∴a(n+1)=3an+2
∴a(n+1) +1 = 3(an+1)
∴(a(n+1) +1 )/(an+1) =3
∴数列{an+1}是等比数列,公比q=3,首项a1 +1 =2
(2)数列{an+1}的前n项和S'n=2(1-3^n)/(1-3) =3^n -1
数列{an}的前n项和Sn=a1+a2+a3+……+an
(a1+1-1)+(a2+1-1)+(a3+1-1)+……+(an+1-1)
=(a1+1)+(a2+1)+(a3+1)+……+(an+1)-n
=3^n -1 -n
=3^n -n-1