若数列{an}满足a2n-a2n-1=p(p为常数,n≥2,n∈N*),则称数列{an}为等方差数列,p为公方差,已知正数等

2025-05-16 08:34:58
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回答(1):

设数列{an}为正数等方差数列,p为公方差,则

a
-
a
=p,
a
-
a
=p
a
-
a
=p
a
-
a
=p

a
-
a
=4p

∵a1=1,∴a2=
1+p
,a5=
1+4p

∵a1,a2,a5成等比数列,
∴1+p=
1+4p

∴p=0或p=2
∵a1≠a2,∴p=2
∴an=
1+2(n-1)
=
2n-1

1
an+an+1
=
1
2n-1
+
2n+1
=
1
2
2n+1
-
2n-1

Tn=
1
a1+a2
+
1
a2+a3
+…+
1
an+an+1
=
1
2
2n+1
-1)
∴A中的整数元素为1,2,3,4,5,6
∵A的非空子集B,若B的元素都是整数,
∴集合A中的完美子集的个数为26-1=63
故选B.