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【拿高分,选好题】高中新课程数学(苏教)二轮复习《专题一 70分填空题大突破与解题技巧》第22题课件

专题一 70分填空题大

突破与解题技法

考查等差数列与等比数列
【例1】? (2012· 苏北四市质量检测)已知等差数列{an},{bn}的 Sn 7n+45 an 前n项和分别为Sn和Tn,若 T = ,且 b 是整数,则n n+3 n 2n 的值为________.

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解析

由题意可知Sn=kn(7n+45),Tn=kn(n+3),所以an

=Sn-Sn-1=kn(7n+45)-k(n-1)(7n+38)=14n+38, b2n=T2n-T2n-1=2kn(2n+3)-k(2n-1)(2n+2)=4n+2,所 an 7n+19 7 1 31 以b = = 2+2 × ∈Z,所以2n+1=31,解得n 2n+1 2n+1 2n =15. 答案 15

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命题趋势1:等差、等比数列基本量的计算,填空题经常考查等 差、等比数列的通项公式、前n项求和、性质等重要内容,考

查运算求解的能力,合理应用等差、等比数列的性质可以简化
运算,所以性质的灵活应用是等差、等比数列的重要考点.

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【突破训练1】 在等差数列{an}中,a1=25,S17=S9,则数列 前n项和最大时,n=________.

解析

因为S17=S9?S17-S9=a10+a11+?+a17=4(a13+a14)

=0,又a1=25>0,所以a13=-a14>0,即该数列前13项是 正数,从第14项开始为负数,所以前13项的和最大. 答案 13

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【例 2】? (2012· 徐州质量检测)在等比数列{an}中,已知 a1+a2 1 =2,a3+a4=1,则 a7+a8+a9+a10 的值为________. 解析 1 等比数列{an}相邻两项的和构成 为首项, 为公比的 2 2

1 等比数列,所以 a7+a8+a9+a10=2×(23+24)=4+8=12. 答案 12

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【突破训练2】 (2012·常州期末)已知等比数列{an}的各项均为

正数,且a1+2a2 =3,a=4a3a7,则数列{an}的通项公式为
________.
解析 a2=4a3a7=4a2,又 4 5 a5 1 an>0,所以 a4=2a5?q=a =2, 4

3 3 ?1?n-1 3 所以 a1+2a2=a1+a1=3?a1=2,所以 an=2×?2? =2n. ? ? 答案 3 an=2n

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【例3】? 等比数列{an}中,a1=2,a8=4,函数f(x)=x(x-a1)(x -a2)?(x-an),则f′(0)的值是________.

解析 由条件得f′(0)=a1a2?a8=(a1a8)4=84=212.
答案 212

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sin2a3-sin2a7 【例4】? 等差数列{an}的公差d∈(0,1),且 =- sin?a3+a7? 1,当n=10时,数列{an}的前n项和Sn取得最小值,则首项 a1的取值范围为________. 解析 sin2a3-sin2a7 因为{an}是等差数列,所以 sin?a3+a7?

?sin a3+sin a7??sin a3-sin a7? = = sin 2a5

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2sin a5cos 2d?-2cos a5sin 2d? π kπ =-sin 4d=-1,得d= 8 + 2 ,k sin 2a5 π ∈Z,又d∈(0,1),所以k=0,即d= .又由S10是{Sn}中的最小 8 9π ? ?a10=a1+ 8 ≤0, 项,所以? ?a11=a1+5π≥0, 4 ? 答案
? 5 9 ? ?- π,- π? 8 ? ? 4

5 9 解得-4π≤a1≤-8π.

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命题趋势2:等差、等比数列与其他知识的综合应用,填空题中 等差数列、等比数列还经常在数列内部,或者与函数、导数、

不等式、恒等式、三角恒等变换等知识综合,构成中高档题,
这类题的训练为数列解答题的求解积累方法和技巧.

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【突破训练 3】 下表给出一个“直角三角形数阵” 1 4 1 1 2,4 3 3 3 4,8,16 ?? 满足每一列成等差数列, 从第三行起, 每一行的数成等比数 列, 且每一行的公比相等, 记第 i 行第 j 列的数为 aij(i≥j, i, j∈N+),则 a83 等于________.
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解析

1 1 由条件得 a81=4+7×4=2,又从第三行起,每一行的数

?1? 1 成等比数列,且每一行的公比相等,都为 ,所以 a83=a81?2?2 2 ? ?

1 1 =2×4=2. 答案 1 2

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