(本小題滿分12分)已知數(shù)列{a
n}的前n項(xiàng)和為S
n, 且滿足條件:4S
n =
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002908365316.gif)
+ 4n – 1 , nÎN*.
(1) 證明:(a
n– 2)
2 –
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002908380331.gif)
="0" (n ³ 2);(2) 滿足條件的數(shù)列不惟一,試至少求出數(shù)列{a
n}的的3個(gè)不同的通項(xiàng)公式 .
(2) 當(dāng)a1 =1且a n + an – 1 = 2時(shí),得an ="1. " 2)當(dāng)a1 =1且a n – a n – 1 =" 2" 時(shí),得an =" 2n–1" .
3)當(dāng)a1 =3且a n – a n – 1 =" 2" 時(shí),得an =" 2n" + 1 . 4)當(dāng)a1 =3且a n + an – 1 = 2時(shí),得an =2(–1)n+ 1 + 1.
(1) 由條件4S
n =
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002908365316.gif)
+ 4n – 1 , nÎN*.得4S
n – 1 =
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002908380331.gif)
+ 4(n – 1 ) – 1,
相減得:4a
n =
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002908365316.gif)
–
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002908380331.gif)
+ 4,化成
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002908365316.gif)
–4a
n+ 4–
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002908380331.gif)
= 0,
∴ (a
n– 2)
2 –
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002908380331.gif)
="0" . 4分
(2) 由(1)得:(a
n –2 + a
n – 1 )(a
n –2 – a
n – 1 ) =" 0∴" a
n + a
n – 1 =" 2 " 或a
n – a
n – 1 =" 2" . 2分
在4S
n =
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002908365316.gif)
+ 4n – 1中,令n = 1,得4a
1 =
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002908536216.gif)
+ 4 – 1,解得:a
1 =1或 a
1 ="3. " 2分
分四種情況:
1)當(dāng)a
1 =1且a
n + a
n – 1 = 2時(shí),得a
n =1.
2)當(dāng)a
1 =1且a
n – a
n – 1 =" 2" 時(shí),得a
n =" 2n–1" .
3)當(dāng)a
1 =3且a
n – a
n – 1 =" 2" 時(shí),得a
n =" 2n" + 1 .
4)當(dāng)a
1 =3且a
n + a
n – 1 = 2時(shí),得a
n =2(–1)
n+ 1 + 1. 每個(gè)1分,有3個(gè)即可
練習(xí)冊(cè)系列答案
相關(guān)習(xí)題
科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:解答題
(本小題滿分13分)
設(shè)數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115118469381.gif)
滿足
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115118500804.gif)
為實(shí)數(shù)
(Ⅰ)證明:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115118515476.gif)
對(duì)任意
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115118547383.gif)
成立的充分必要條件是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115118562314.gif)
;
(Ⅱ)設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115118578343.gif)
,證明:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115118609620.gif)
;
(Ⅲ)設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115118578343.gif)
,證明:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115118687856.gif)
查看答案和解析>>
科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:解答題
(本題滿分15分)已知分別以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115606430372.gif)
為公差的等差數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115606461278.gif)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115606476278.gif)
,滿足
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115606492555.gif)
.(Ⅰ)若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115606523253.gif)
,且存在正整數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115606554204.gif)
,使得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115606617644.gif)
,求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115606632212.gif)
的最小值;(Ⅱ)若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115606648364.gif)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115606742450.gif)
且數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115606773757.gif)
,的前項(xiàng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115606804192.gif)
和
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115606835220.gif)
滿足
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115606851664.gif)
,求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115606461278.gif)
的通項(xiàng)公式.
查看答案和解析>>
科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:解答題
(本小題滿分14分)
設(shè)數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115017551381.gif)
滿足
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115017582244.gif)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115017597258.gif)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115017644823.gif)
.?dāng)?shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115017691385.gif)
滿足
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115017722245.gif)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115017738486.gif)
是非零整數(shù),且對(duì)任意的正整數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115017769204.gif)
和自然數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115017785199.gif)
,都有
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115017816649.gif)
.
(1)求數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115017551381.gif)
和
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115017691385.gif)
的通項(xiàng)公式;
(2)記
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115017925615.gif)
,求數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115017956270.gif)
的前
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115017987192.gif)
項(xiàng)和
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823115018065220.gif)
.
查看答案和解析>>
科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:解答題
(本題滿分16滿分)設(shè)正項(xiàng)數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002928801381.gif)
的前
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002928816192.gif)
項(xiàng)和為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002928832220.gif)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002928848199.gif)
為非零常數(shù).已知對(duì)任意正整數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002928863348.gif)
,當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002928879358.gif)
時(shí),
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002928879529.gif)
總成立.
(1)證明:數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002928801381.gif)
是等比數(shù)列;(2) 若正整數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002928926387.gif)
成等差數(shù)列,求證:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002928941452.gif)
≥
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002928957371.gif)
.
查看答案和解析>>
科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:解答題
(本題滿分14分)
已知數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002851641270.gif)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002851657271.gif)
滿足
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002851673249.gif)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002851688254.gif)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002851688649.gif)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002851704442.gif)
。
(1)求數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002851719270.gif)
的通項(xiàng)公式;
(2)求數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002851735381.gif)
的通項(xiàng)公式;
(3)數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002851751259.gif)
滿足
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002851766548.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002851782431.gif)
,求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002851813764.gif)
。
查看答案和解析>>
科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:解答題
(本題滿分12分)已知數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002815699264.gif)
的各項(xiàng)均是正數(shù),其前
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002815715189.gif)
項(xiàng)和為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002815886203.gif)
,滿足
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002815902511.gif)
,其中
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002815917200.gif)
為正常數(shù),且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002815949240.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002815964332.jpg)
(1)求數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002815699264.gif)
的通項(xiàng)公式;(2)設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002816105694.gif)
,數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002816229428.gif)
的前
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002815715189.gif)
項(xiàng)和為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002816276198.gif)
,求證:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002816292388.gif)
查看答案和解析>>
科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:單選題
一個(gè)三角形的三個(gè)內(nèi)角A、B、C成等差數(shù)列,那么tan(A+C)的值是( �。�
查看答案和解析>>
科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:單選題
等比數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002735233381.gif)
的前n項(xiàng)和為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002735248202.gif)
,且4
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002735264206.gif)
,2
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002735295210.gif)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002735311209.gif)
成等差數(shù)列。若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002735264206.gif)
=1,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823002735342201.gif)
="( " )
查看答案和解析>>