wwweinvmxcom:等比数列通项、求和典型题

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等比数列的性质以及常见题型 一 公比 q 的运用 1.在等比数列 ?an ? 中, a2 ? 2 , a5 ? 128 ,则它的公比 q ? _______,前 n 项和 S n ? _______. 2.在等比数列 {an } 中, a1 ? ?16 , a4 ? 8 ,则 a7 ? ( 3.在等比数列 ?an ? 中, a2010 ? 8a2007 ,则公比 q 的值为( 4.等比数列 ?an ? 的公比为 2 ,则
2a1 ? a2 的值为 2a3 ? a4

)A. ?4 )A.2

B. ?4 B. 3

C. ?2 C.4

D. ?2 D. 8

5.设等比数列 ?an ? 的前 n 项和为 S n ,若

S6 S ? 3 ,则 9 ? ( S3 S6

)A. 2

B.

7 3

C.

8 3

D. 3
. )

6.等比数列 ?an ? 的首项 a1 ? ?1 ,前 n 项和为 S n ,公比 q ? 1 ,若

S10 a 31 = ,则 10 等于 32 S5 a5

7.在等比数列 {an } 中,公比 q ? 2 ,且 a1 ? a2 ? a3 ? ?? a30 ? 230 ,则 a3 ? a6 ? a9 ? ?? a30 等于( A. 210 二 B. 220 C. 216 D. 215

m ? n ? s ? t ? aman ? as at 性质的应用


1.在各项均为正数的等比数列 ?bn ? 中,若 b7 ? b8 ? 3 ,则 log3 b1 ? log3 b2 ? …… ? log3 b14 等于( A. 5 B. 6 C. 7 D. 8
log3 a10 ? (

2.等比数列 {an } 的各项均为正数,且 a5 a6 ? a4 a7 ? 18 ,则 log3 a1 ? log3 a2 ? A.12 B.10 C.8

)

D. 2 ? log3 5 .

3.在等比数列 ?an ? 中,若 a3 , a9 是方程 3x2 ? 11x ? 9 ? 0 的两根,则 a6 的值是 4.在等比数列中

an>0, a3a8 ? 27 , log3a1 ? log3a2 ? log3a3 ?

? log3a10 ? ________。

5.已知各项均为正数的等比数列 ?an ? , a1a2 a3 ? 5 , a7 a8 a9 ? 10 ,则 a4 a5 a6 ? A. 5 2 B.7 C.6 D. 4 2

6.设 ?an ? 是任意等比数列,它的前 n 项和,前 2n 项和与前 3n 项和分别为 X , Y , Z ,则下列等式中恒成立的是 A. X ? Z ? 2Y C. Y 2 ? XZ 三 求数列 an 的通项公式 B. Y (Y ? X ) ? Z (Z ? X ) D. Y (Y ? X ) ? X (Z ? X )

(一)构造法求 an 构造法 (形式为: an?1 ? kan ? b, k为常数, b不一定为常数,可能与 n有关) 1.已知数列 ?an ? 中,a 1 =3,a n?1 =

1 ? a n +1(n∈N )求数列 ?an ? 的通项公式 2

2.已知数列 ?an ? 中,a 1 =1,a n?1 =3a n +2,求数列 ?an ? 的通项公式

(二)根据题意构造 1.数列{an}的前 n 项和记为 Sn,已知 a1=1,an+1= (1)求证:数列{
Sn }是等比数列. n n?2 Sn(n=1,2,3…). n

(2)求数列 ?a n ?的通项公式.

2.已知在数列 ?a n ?中, an?1 ? 2an ? n ? 1 , a1 ? 1 , (1)证明数列 ?an ? n? 为等比数列 (2)求数列 ?a n ?的通项公式。

2.裂项相减(等差与等比之积的形式 an 为等差数列, bn 为等比数列,则数列 anb n 的前 n 项和) 例题:设 {an } 是等差数列, {bn } 是各项都为正数的等比数列,且 a1 ? b1 ? 1 , a3 ? b5 ? 21 , a5 ? b3 ? 13 (Ⅰ)求 {an } , {bn } 的通项公式; (Ⅱ)求数列 ?

? an ? ? 的前 n 项和 Sn . ? bn ?



其他类型

1.已知数列 ?an ? 的前 n 项和是 S n ,且 2S n ? an ? 1 (n ? N * ) . (Ⅰ) 求证 :数列 ?an ? 是等比数列; (Ⅱ) 记 bn ? 10 ? log9 an ,求 ?bn ? 的前 n 项和 Tn 的最大值及相应的 n 值.


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