(本小題滿分16分)
已知數(shù)列

滿足

+

=4n-3(n∈

).
(1)若數(shù)列

是等差數(shù)列,求

的值;
(2)當

=2時,求數(shù)列

的前n項和

;
(3)若對任意n∈

,都有

≥5成立,求

的取值范圍.
解析:(1)若數(shù)列

是等差數(shù)列,則

=

+(n-1)d,

=

+nd.
由

+

=4n-3,得(

+nd)+[

+(n-1)d]=4n-3,即2d=4,

-d=-3,解得d=2,

=

.
(2)由

+

=4n-3(n∈

),得

+

=4n+1(n∈

).
兩式相減,得

-

=4.
所以數(shù)列

是首項為

,公差為4的等差數(shù)列.
數(shù)列

是首項為

,公差為4的等差數(shù)列.
由

+

=1,

=2,得

=-1.
所以

=

(k∈Z).
①當n為奇數(shù)時,

=2n,

=2n-3.

=

+

+

+…+

=(

+

)+(

+

)+…+(

+

)+

=1+9+…+(4n-11)+2n=

+2n=

.
②當n為偶數(shù)時,

=

+

+

+…+

=(

+

)+(

+

)+…+(

+

)==1+9+…+(4n-7) =

.
所以

=

(k∈Z).
(3)由(2)知,

=

(k∈Z).
①當n為奇數(shù)時,

=2n-2+

,

=2n-1-

.
由

≥5,得

-

≥

+16n-10.
令

=

+16n-10=

+6.
當n=1或n=3時,

=2,所以

-

≥2.
解得

≥2或

≤-1.
②當n為偶數(shù)時,

=2n-3-

,

=2n+

.
由

≥5,得

+

≥

+16n-12.
令

=

+16n-12=

+4.
當n=2時,

=4,所以

+

≥4.
解得

≥1或

≤-4.
綜上所述,

的取值范圍是

,

,

.
練習冊系列答案
相關習題
科目:高中數(shù)學
來源:不詳
題型:解答題
(本小題滿分12分)
已知數(shù)列

中,

,

,其前

項和為

,且當

時,

.
(Ⅰ)求證:數(shù)列

是等比數(shù)列;
(Ⅱ)求數(shù)列

的通項公式;
(Ⅲ)令

,記數(shù)列

的前

項和為

,證明對于任意的正整數(shù)

,都有

成立.
查看答案和解析>>
科目:高中數(shù)學
來源:不詳
題型:單選題
已知數(shù)列{a
n}中,a
n=(n∈N
*),則在數(shù)列{a
n}的前50項中最小項和最大項分別是( )
A.a(chǎn)1,a50 | B.a(chǎn)1,a8 |
C.a(chǎn)8,a9 | D.a(chǎn)9,a50 |
查看答案和解析>>
科目:高中數(shù)學
來源:不詳
題型:解答題
已知等差數(shù)列

的前

項和為

,且


.
(Ⅰ)求數(shù)列

的通項公式;
(Ⅱ)求證:

查看答案和解析>>
科目:高中數(shù)學
來源:不詳
題型:單選題
已知

是等差數(shù)列,

是其前n項和,

,

,則過點P(3,

),Q(4,

)的直線的斜率是
A.4 | B. | C.-4 | D.-14 |
查看答案和解析>>
科目:高中數(shù)學
來源:不詳
題型:填空題
已知數(shù)列{

}的前

項和為

=

,則它的通項公式為
查看答案和解析>>
科目:高中數(shù)學
來源:不詳
題型:填空題
.數(shù)列

的前

項和為

,且

,則

________________
查看答案和解析>>
科目:高中數(shù)學
來源:不詳
題型:單選題
已知等差數(shù)列

滿足

,則

等于( )
A.10 | B.8 | C.6 | D.4 |
查看答案和解析>>