設滿足以下兩個條件得有窮數(shù)列
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為
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階“期待數(shù)列”:
①
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,②
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.
(1)若等比數(shù)列
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為
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階“期待數(shù)列”,求公比
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;
(2)若一個等差數(shù)列
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既為
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階“期待數(shù)列”又是遞增數(shù)列,求該數(shù)列的通項公式;
(3)記
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階“期待數(shù)列”
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的前
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項和為
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.
(
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)求證:
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;
(
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)若存在
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,使
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,試問數(shù)列
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
是否為
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階“期待數(shù)列”?若能,求出所有這樣的數(shù)列;若不能,請說明理由.
(1)
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;(2)
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;(3)(
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)證明見解析;(
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)不能,理由見解析.
試題分析:
(1)由
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階“期待數(shù)列”定義,當
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,結合已知條件①求得等比數(shù)列的公比
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,若
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,由①得,
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,得
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,不可能,所以
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;
(2)設出等差數(shù)列的公差,結合①②求出公差,再由前
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項和為
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求出首項,則等差數(shù)列的通項公式可求;
(3)(
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)由
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階“期待數(shù)列”
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前
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項中所有的和為0,所有項的絕對值之和為1,求得所有非負項的和為
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,所有負項的和為
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,從而得到答案;
(
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)借助于(
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)中結論知,數(shù)列
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的前
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項和為
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,且滿足
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,再由
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,得到
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,從而說明
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與
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不能同時成立.
(1) 若
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,則由①
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由
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,所以
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,得
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,
由②得
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或
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,滿足題意.
若
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,由①得,
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,得
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,不可能.
綜上所述
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.
(2)設等差數(shù)列
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的公差為
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.
因為
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,所以
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.
所以
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.
因為
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,所以由
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,得
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.
由題中的①、②得
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,
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,
兩式相減得
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, 即
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. 又
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,得
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.
所以
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.
(3) 記
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中非負項和為
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,負項和為
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.
則
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, 得
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.
(
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) 因為
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,所以
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.
(
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) 若存在
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,使
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,由前面的證明過程知:
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,
且
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.
記數(shù)列
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的前
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項和為

.若
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為
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階“期待數(shù)列”,
則由(
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)知,
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. 所以
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因為
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, 所以

.
所以
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,

.
又
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, 則
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.
所以
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.
所以
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與
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不能同時成立.
所以對于有窮數(shù)列
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
,若存在
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,使
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,
則數(shù)列
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不能為
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階“期待數(shù)列”.
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相關習題
科目:高中數(shù)學
來源:不詳
題型:解答題
設各項均為正數(shù)的數(shù)列
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的前
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項和為
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,滿足
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,且
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恰為等比數(shù)列
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的前三項.
(1)證明:數(shù)列
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為等差數(shù)列; (2)求數(shù)列
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的前
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項和
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.
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科目:高中數(shù)學
來源:不詳
題型:解答題
數(shù)列
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的首項
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,
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求數(shù)列
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的通項公式;
設
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的前
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項和為
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,若
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的最小值為
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,求
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的取值范圍?
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科目:高中數(shù)學
來源:不詳
題型:解答題
(2013·杭州模擬)已知數(shù)列{a
n}的前n項和S
n=-a
n-
n-1+2(n∈N
*),數(shù)列{b
n}滿足b
n=2
na
n.
(1)求證數(shù)列{b
n}是等差數(shù)列,并求數(shù)列{a
n}的通項公式.
(2)設數(shù)列
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的前n項和為T
n,證明:n∈N
*且n≥3時,T
n>

.
(3)設數(shù)列{c
n}滿足a
n(c
n-3
n)=(-1)
n-1λn(λ為非零常數(shù),n∈N
*),問是否存在整數(shù)λ,使得對任意n∈N
*,都有c
n+1>c
n.
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科目:高中數(shù)學
來源:不詳
題型:解答題
數(shù)列
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是等差數(shù)列,
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,前四項和
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。
(1)求數(shù)列
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的通項公式;
(2)記
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,計算
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。
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科目:高中數(shù)學
來源:不詳
題型:解答題
(2013•重慶)設數(shù)列{an}滿足:a1=1,an+1=3an,n∈N+.
(1)求{an}的通項公式及前n項和Sn;
(2)已知{bn}是等差數(shù)列,Tn為前n項和,且b1=a2,b3=a1+a2+a3,求T20.
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科目:高中數(shù)學
來源:不詳
題型:解答題
(2011•浙江)已知公差不為0的等差數(shù)列{a
n}的首項a
1為a(a∈R)設數(shù)列的前n項和為S
n,且
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,
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,

成等比數(shù)列.
(1)求數(shù)列{a
n}的通項公式及S
n;
(2)記A
n=
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+
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+
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+…+
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,B
n=
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+
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+…+

,當n≥2時,試比較A
n與B
n的大�。�
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科目:高中數(shù)學
來源:不詳
題型:單選題
(2014·佛山模擬)數(shù)列{a
n}滿足a
n+a
n+1=
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(n∈N
*),a
2=2,S
n是數(shù)列{a
n}的前n項和,則S
2015為( )
A.502 | B.504 | C. | D.2015 |
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科目:高中數(shù)學
來源:不詳
題型:單選題
設數(shù)列
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,則對任意正整數(shù)
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都成立的是( )
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