已知数列{an}满足a1=1,an=3n-1+an-1(n≥2).(Ⅰ)求a2,a3;(Ⅱ)证明an=3n?12

2025-02-26 23:19:35
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回答1:

(Ⅰ)∵a1=1,
∴a2=3+1=4,
∴a3=32+4=13;

(Ⅱ)证明:由已知an-an-1=3n-1,n≥2
故an=(an-an-1)+(an-1-an-2)+…+(a2-a1)+a1
=3n?1+3n?2+…+3+1=
3n?1
2
.n≥2
当n=1时,也满足上式.
所以an
3n?1
2