题目内容
在数列{an}中,a1=1,an+1•
=8
(Ⅰ)求a2,a3;
(Ⅱ)设bn=log2an,求证:{bn-2}为等比数列;
(Ⅲ)求{an}的前n项积Tn.
an |
(Ⅰ)求a2,a3;
(Ⅱ)设bn=log2an,求证:{bn-2}为等比数列;
(Ⅲ)求{an}的前n项积Tn.
(Ⅰ)∵a2•
=8,a1=1,
∴a2=8.
∵a3•
=8,a1=8,
∴a3=2
.
(Ⅱ)证明:∵
=
=
=
═
×
=-
.
∴{bn-2}为等比数列,首项为b1-2,即为-2,其公比为-
.
(Ⅲ)设数列{bn-2}的前n项和为Sn
.
∴log2Tn=
[(-
)n-1]+2n,
∴Tn=2
[(-
)n-1]+2n.
a1 |
∴a2=8.
∵a3•
a2 |
∴a3=2
2 |
(Ⅱ)证明:∵
bn+1-2 |
bn-2 |
log2an+1-2 |
log2an-2 |
=
log2
| ||||
log2an-2 |
3-
| ||
log2an-2 |
═
1 |
2 |
2-log2an |
log2an-2 |
1 |
2 |
∴{bn-2}为等比数列,首项为b1-2,即为-2,其公比为-
1 |
2 |
(Ⅲ)设数列{bn-2}的前n项和为Sn
|
∴log2Tn=
4 |
3 |
1 |
2 |
∴Tn=2
4 |
3 |
1 |
2 |
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