已知数列{an}和{bn}满足条件:a1=3,a2=2,b1=b2=2,b3=3,且数列{an-1}为等比数列,数列{bn+1-bn}为等差

2025-05-13 15:35:59
推荐回答(1个)
回答(1):

(Ⅰ)解:∵a1=3,a2=2,数列{an-1}为等比数列,
∴an-1=2?(

1
2
)n?1=22-n
∴an=22-n+1,
∵b1=b2=2,b3=3,数列{bn+1-bn}为等差数列,
∴bn+1-bn=n-1,
∴bn=b1+(b2-b1)+…+(bn-bn-1)=
n2?n+4
2

(Ⅱ)证明:n≥3时,
1
bn?2
=
2
n(n?1)
=2(
1
n?1
-
1
n
),
1
b3?2
+
1
b4?2
+…+
1
bn?2
=2(
1
2
-
1
3
+…+
1
n?1
-
1
n
)=2(
1
2
-
1
n
)≤
1
3
<2.