【单词记忆法】2015《三维设计》高考英语(江苏专用)大一轮复习—配套高效记单词课件：2构词记忆法
《三维设计》2015届高考数学苏教,理科大一轮复习配套讲义备考基础查清热点命题悟通：第五章 数 列必记知
第五章数列第一节数列的概念与简单表示法对应学生用书P671．数列的定义、分类与通项公式(1)数列的定义：①数列：按照一定顺序排列的一列数．②数列的项：数列中的每一个数．(2)数列的分类：分类标准类型满足条件有穷数列项数有限项数无穷数列项数无限(3)如果数列{a n }的第n 项与序号n 之间的关系可以用一个式子来表示，那么这个公式叫做这个数列的通项公式．2．数列的递推公式如果已知数列{a n }的首项(或前几项)，且任一项a n 与它的前一项a n －1(n ≥2)(或前几项)间的关系可用一个公式来表示，那么这个公式叫数列的递推公式．1．数列是按一定“次序”排列的一列数，一个数列不仅与构成它的“数”有关，而且还与这些“数”的排列顺序有关．2．易混项与项数两个不同的概念，数列的项是指数列中某一确定的数，而项数是指数列的项对应的位置序号．[试一试]1．已知数列{a n }的前4项为1,3,7,15，写出数列{a n }的一个通项公式为________． 答案：a n ＝2n －1(n ∈N *)2．已知数列{a n }的通项公式是a n ＝⎩⎪⎨⎪⎧2·3n －1(n 为偶数)，2n －5(n 为奇数)，则a 4·a 3＝________.解析：a 4·a 3＝2×33·(2×3－5)＝54. 答案：541．辨明数列与函数的关系数列是一种特殊的函数，即数列是一个定义在非零自然数集或其子集上的函数，当自变量依次从小到大取值时所对应的一列函数值，就是数列．2．明确a n 与S n 的关系a n ＝⎩⎪⎨⎪⎧S 1 (n ＝1)，S n －S n －1 (n ≥2).[练一练]1．(2013·南京、淮安二模)已知数列{a n }的通项为a n ＝7n ＋2，数列{b n }的通项为b n ＝n 2.若将数列{a n }，{b n }中相同的项按从小到大的顺序排列后记作数列{c n }，则c 9的值是________．解析：法一：由a n ＝7n ＋2，b n＝n 2列出部分项得⎩⎪⎨⎪⎧a 1＝9，b 3＝9，⎩⎪⎨⎪⎧a 2＝16，b 4＝16，⎩⎪⎨⎪⎧a 14＝100，b 10＝100，⎩⎪⎨⎪⎧a 17＝121，b 11＝121，⎩⎪⎨⎪⎧a 41＝289，b 17＝289，⎩⎪⎨⎪⎧a 46＝324，b 18＝324，易发现在数列{b n }中符合条件的数呈周期变化，且周期为7.每个周期内第3,4个数符合题意，故c 9在第5个周期的第3个数，即c 9＝(4×7＋3)2＝312＝961.法二：令a n ＝b m ，则7n ＋2＝m 2，即7(n －1)＝(m －3)(m ＋3)．易知m ＋3或m －3是7的整数倍，所以当m ＝3,4,10,11,17,18,24,25,31,32，…时满足等式，故c 9＝312＝961.答案：9612．(2014·苏锡常镇调研)设u (n )表示正整数n 的个位数，a n ＝u (n 2)－u (n )，则数列{a n }的前2 014项和等于________．解析：因为n 与n ＋10的个位数字相同且周期为10，又a 1＝0，a 2＝4－2＝2，a 3＝9－3＝6，a 4＝6－4＝2，a 5＝5－5＝0，a 6＝6－6＝0，a 7＝9－7＝2，a 8＝4－8＝－4，a 9＝1－9＝－8，a 10＝0，所以a 1＋a 2＋…＋a 10＝0，即a 1＋a 2＋…＋a 2 014＝a 1＋a 2＋a 3＋a 4＝10.答案：10对应学生用书P67考点一由数列的前几项求数列的通项公式1.(2014· 2 4 6 8 10 12 14 16 18 20 …则第n (n ≥4)行从左向右的第4个数为________．解析：从数表可知，所有的数是由偶数组成的，第n 行有n 个偶数，从而前n －1行有1＋2＋…＋(n －1)＝n (n －1)2个偶数，第(n ≥4)行从左向右的第4个数是第n (n －1)2＋4个偶数，所以是n 2－n ＋8.答案：n 2－n ＋82．根据数列的前几项，写出各数列的一个通项公式： (1)4,6,8,10，…；(2)－11×2，12×3，－13×4，14×5，…； (3)a ，b ，a ，b ，a ，b ，…(其中a ，b 为实数)； (4)9,99,999,9 999，….解：(1)各数都是偶数，且最小为4，所以通项公式a n ＝2(n ＋1)(n ∈N *)．(2)这个数列的前4项的绝对值都等于序号与序号加1的积的倒数，且奇数项为负，偶数项为正，所以它的一个通项公式a n ＝(－1)n ×1n (n ＋1).(3)这是一个摆动数列，奇数项是a ，偶数项是b ，所以此数列的一个通项公式a n ＝⎩⎪⎨⎪⎧a ，n 为奇数，b ，n 为偶数.(4)这个数列的前4项可以写成10－1,100－1,1 000－1，10 000－1，所以它的一个通项公式a n ＝10n －1.[备课札记] [类题通法]用观察法求数列的通项公式的技巧(1)根据数列的前几项求它的一个通项公式，要注意观察每一项的特点，观察出项与n 之间的关系、规律，可使用添项、通分、分割等办法，转化为一些常见数列的通项公式来求．对于正负符号变化，可用(－1)n 或(－1)n＋1来调整．(2)根据数列的前几项写出数列的一个通项公式是不完全归纳法，它蕴含着“从特殊到一般”的思想．[n n n (1)S n ＝2n 2－3n ； (2)S n ＝3n ＋b .[解] (1)a 1＝S 1＝2－3＝－1，当n ≥2时，a n ＝S n －S n －1＝(2n 2－3n )－[2(n －1)2－3(n －1)]＝4n －5，由于a 1也适合此等式，∴a n ＝4n －5. (2)a 1＝S 1＝3＋b ， 当n ≥2时，a n ＝S n －S n －1 ＝(3n ＋b )－(3n －1＋b )＝2·3n －1. 当b ＝－1时，a 1适合此等式． 当b ≠－1时，a 1不适合此等式． ∴当b ＝－1时，a n ＝2·3n －1；当b ≠－1时，a n ＝⎩⎪⎨⎪⎧3＋b ，n ＝1，2·3n －1，n ≥2.[备课札记] [类题通法]已知数列{a n }的前n 项和S n ，求数列的通项公式，其求解过程分为三步：(1)先利用a 1＝S 1求出a 1；(2)用n －1替换S n 中的n 得到一个新的关系，利用a n ＝S n －S n －1(n ≥2)便可求出当n ≥2时a n 的表达式；(3)对n ＝1时的结果进行检验，看是否符合n ≥2时a n 的表达式，如果符合，则可以把数列的通项公式合写；如果不符合，则应该分n ＝1与n ≥2两段来写．[针对训练]已知各项均为正数的数列{a n }的前n 项和满足S n >1，且6S n ＝(a n ＋1)(a n ＋2)，n ∈N *，求{a n }的通项公式．解：由a 1＝S 1＝16(a 1＋1)(a 1＋2)，解得a 1＝1或a 1＝2， 由已知a 1＝S 1>1，因此a 1＝2.又由a n ＋1＝S n ＋1－S n ＝16(a n ＋1＋1)(a n ＋1＋2)－16(a n ＋1)·(a n ＋2)，得a n ＋1－a n －3＝0或a n ＋1＝－a n . 因为a n >0，故a n ＋1＝－a n 不成立，舍去． 因此a n ＋1－a n －3＝0.即a n ＋1－a n ＝3，从而{a n }是以公差为3，首项为2的等差数列，故{a n }的通项公式为a n＝3n －1.递推公式和通项公式是数列的两种表示方法，它们都可以确定数列中的任意一项，只是由递推公式确定数列中的项时，不如通项公式直接，归纳起来常见的命题角度有：(1)形如a n ＋1＝a n f (n )，求a n ； (2)形如a n ＋1＝a n ＋f (n )，求a n ；(3)形如a n ＋1＝Aa n ＋B (A ≠0且A ≠1)，求a n . 角度一 形如a n ＋1＝a n f (n )，求a n1．已知数列{a n }中，a 1＝1，前n 项和S n ＝n ＋23a n .(1)求a 2，a 3； (2)求{a n }的通项公式．解：(1)由S 2＝43a 2得3(a 1＋a 2)＝4a 2，解得a 2＝3a 1＝3.由S 3＝53a 3得3(a 1＋a 2＋a 3)＝5a 3，解得a 3＝32(a 1＋a 2)＝6.(2)由题设知a 1＝1. 当n ≥2时，有a n ＝S n －S n －1＝n ＋23a n －n ＋13a n －1，整理得a n ＝n ＋1n －1a n －1.即a n a n －1＝n ＋1n －1. ∴a n ＝a 1·a 2a 1·a 3a 2·a 4a 3·a 5a 4·…·a n －2a n －3·a n －1a n －2·a n a n －1＝1·31·42·53·64·…·n －1n －3·n n －2·n ＋1n －1＝n (n ＋1)2(n ≥2)当n ＝1时，a 1＝1.综上可知，{a n }的通项公式a n ＝n (n ＋1)2.角度二 形如a n ＋1＝a n ＋f (n )，求a n 2．已知a 1＝2，a n ＋1＝a n ＋3n ＋2，求a n . 解：∵a n ＋1－a n ＝3n ＋2， ∴a n －a n －1＝3n －1(n ≥2)，∴a n ＝(a n －a n －1)＋(a n －1－a n －2)＋…＋(a 2－a 1)＋a 1＝n (3n ＋1)2(n ≥2)．当n ＝1时，a 1＝12×(3×1＋1)＝2符合公式，∴a n ＝32n 2＋n 2.角度三 形如a n ＋1＝Aa n ＋B (A ≠0且A ≠1)，求a n 3．已知数列{a n }满足a 1＝1，a n ＋1＝3a n ＋2，求a n . 解：∵a n ＋1＝3a n ＋2，∴a n ＋1＋1＝3(a n ＋1)，∴a n ＋1＋1a n ＋1＝3，∴数列{a n ＋1}为等比数列，公比q ＝3， 又a 1＋1＝2，∴a n ＋1＝2·3n －1，∴a n ＝2·3n －1－1.[备课札记] [类题通法]由数列的递推公式求通项公式时，若递推关系为a n ＋1＝a n ＋f (n )或a n ＋1＝f (n )·a n ，则可以分别通过累加、累乘法求得通项公式，另外，通过迭代法也可以求得上面两类数列的通项公式，(如角度二)，注意：有的问题也可利用构造法，即通过对递推式的等价变形，(如角度三)转化为特殊数列求通项．对应学生用书P69[课堂练通考点]1．(2014·苏北四市质检)在数列{a n }中，已知a 1＝2，a 2＝3，当n ≥2时，a n ＋1是a n ·a n －1的个位数，则a 2 014＝________.解析：由题意，该数列除前2项外，从第3项往后是周期为6的周期数列，故a 2 014＝a 4＝8.答案：82．(2013·盐城三调)已知函数f (x )＝⎩⎪⎨⎪⎧(3－a )x －3，x ≤7，a x －6， x >7，数列{a n }满足a n ＝f (n )，n ∈N *，且数列{a n }是递增数列，则实数a 的取值范围是________．解析：由题意可知⎩⎪⎨⎪⎧3－a >0，a >1，f (8)>f (7)，解得a ∈(2,3)．答案(2,3)3．已知数列{a n }满足a st ＝a s a t (s ，t ∈N *)，且a 2＝2，则a 8＝________. 解析：令s ＝t ＝2，则a 4＝a 2×a 2＝4，令s ＝2，t ＝4，则a 8＝a 2×a 4＝8. 答案：84．已知数列{a n }中，a 1＝1，a n ＋1＝(－1)n (a n ＋1)，记S n 为{a n }前n 项的和，则S 2 013＝____________.解析：由a 1＝1，a n ＋1＝(－1)n (a n ＋1)可得该数列是周期为4的数列，且a 1＝1，a 2＝－2，a 3＝－1，a 4＝0.所以S 2 013＝503(a 1＋a 2＋a 3＋a 4)＋a 2 013＝503×(－2)＋1＝－1 005.答案：－1 0055．已知数列{a n }的前n 项和S n ＝2n 2＋2n ，数列{b n }的前n 项和T n ＝2－b n .求数列{a n }与{b n }的通项公式．解：∵当n ≥2时，a n ＝S n －S n －1＝(2n 2＋2n )－[2(n －1)2＋2(n －1)]＝4n ， 当n ＝1时，a 1＝S 1＝4也适合， ∴{a n }的通项公式是a n ＝4n (n ∈N *)． ∵T n ＝2－b n ，∴当n ＝1时，b 1＝2－b 1，b 1＝1.当n ≥2时，b n ＝T n －T n －1＝(2－b n )－(2－b n －1)， ∴2b n ＝b n －1.∴数列{b n }是公比为12，首项为1的等比数列．∴b n ＝⎝⎛⎭⎫12n －1.[课下提升考能]第Ⅰ组：全员必做题1．(2013·盐城二调)数列{a n }满足a n ＋a n －1＝12(n ∈N *)，a 1＝1，S n 是{a n }的前n 项和，则S 21＝________.解析：这个数列为“等和数列”，分别计算数列的前几项可以发现该数列为周期数列，周期为2.故S 21＝(1－12)×10＋1＝6.答案：62．已知数列{a n }的前n 项和为S n ，且S n ＝2(a n －1)，则a 2等于________． 解析：由题可知S n ＝2(a n －1)， 所以S 1＝a 1＝2(a 1－1)，解得a 1＝2.又S 2＝a 1＋a 2＝2(a 2－1)，解得a 2＝a 1＋2＝4. 答案：43．设数列{a n }满足：a 1＝2，a n ＋1＝1－1a n，记数列{a n }的前n 项之积为T r ，则T 2 013的值为________．解析：由a 2＝12，a 3＝－1，a 4＝2可知，数列{a n }是周期为3的周期数列，从而T 2 013＝(－1)671＝－1.答案：－14．若数列{a n }满足：a 1＝19，a n ＋1＝a n －3(n ∈N *)，则数列{a n }的前n 项和数值最大时，n 的值为________．解析：∵a 1＝19，a n ＋1－a n ＝－3，∴数列{a n }是以19为首项，－3为公差的等差数列， ∴a n ＝19＋(n －1)×(－3)＝22－3n . 设{a n }的前k 项和数值最大，则有⎩⎪⎨⎪⎧a k ≥0，a k ＋1≤0，k ∈N *，∴⎩⎪⎨⎪⎧22－3k ≥0，22－3(k ＋1)≤0， ∴193≤k ≤223， ∵k ∈N *，∴k ＝7. ∴满足条件的n 的值为7. 答案：75．已知数列{a n }的前n 项和S n ＝2a n －1，则满足a nn ≤2的正整数n 的集合为________．解析：因为S n ＝2a n －1，所以当n ≥2时，S n －1＝2a n －1－1， 两式相减得a n ＝2a n －2a n －1， 整理得a n ＝2a n －1，所以{a n }是公比为2的等比数列， 又因为a 1＝2a 1－1， 解得a 1＝1，故{a n }的通项公式为a n ＝2n －1. 而a nn ≤2，即2n －1≤2n ， 所以有n ＝1,2,3,4. 答案：{1,2,3,4}6．在数列－1,0，19，18，…，n －2n 2，…中，0.08是它的第____________项．解析：令n －2n 2＝0.08，得2n 2－25n ＋50＝0，即(2n －5)(n －10)＝0.解得n ＝10或n ＝52(舍去)．∴a 10＝0.08.答案：107．已知数列{a n }的前n 项和S n ＝2n －3，则数列{a n }的通项公式为________．解析：当n ≥2时，a n ＝S n －S n －1＝2n －1，当n ＝1时，a 1＝S 1＝－1，所以a n ＝⎩⎪⎨⎪⎧－1，n ＝1，2n －1，n ≥2.答案：a n ＝⎩⎪⎨⎪⎧－1，n ＝1，2n －1，n ≥28．数列{a n }满足：a 1＋3a 2＋5a 3＋…＋(2n －1)·a n ＝(n －1)·3n ＋1＋3(n ∈N *)，则数列{a n }的通项公式a n ＝________.解析：a 1＋3a 2＋5a 3＋…＋(2n －3)·a n －1＋(2n －1)·a n ＝(n －1)·3n ＋1＋3，把n 换成n －1得，a 1＋3a 2＋5a 3＋…＋(2n －3)·a n －1＝(n －2)·3n ＋3，两式相减得a n ＝3n .答案：3n9．已知数列{a n }满足a 1＝1，a 2＝2，且a n ＝a n －1a n －2(n ≥3)，则a 2 014＝________.解析：将a 1＝1，a 2＝2代入a n ＝a n －1a n －2得a 3＝a 2a 1＝2，同理可得a 4＝1，a 5＝12，a 6＝12，a 7＝1，a 8＝2，故数列{a n }是周期为6的周期数列，故a 2 014＝a 335×6＋4＝a 4＝1.答案：110．已知数列{a n }的通项公式为a n ＝n 2－21n ＋20. (1)n 为何值时，a n 有最小值？并求出最小值； (2)n 为何值时，该数列的前n 项和最小？解：(1)因为a n ＝n 2－21n ＋20＝⎝⎛⎭⎫n －2122－3614，可知对称轴方程为n ＝212＝10.5.又因n ∈N *，故n ＝10或n ＝11时，a n 有最小值，其最小值为112－21×11＋20＝－90.(2)设数列的前n 项和最小，则有a n ≤0，由n 2－21n ＋20≤0，解得1≤n ≤20，故数列{a n }从第21项开始为正数，所以该数列的前19或20项和最小．第Ⅱ组：重点选做题1．(2014·南通期末)在数列{a n }中，a 1＝6且a n －a n －1＝a n －1n ＋n ＋1(n ∈N *，n ≥2)，则这个数列的通项公式a n ＝________.解析：法一：由题意得a 1＝6，a 2＝12，a 3＝20，a 4＝30，…由此猜想出a n ＝(n ＋1)(n ＋2)．法二：由题意得a n n ＋1＝a n －1n ＋1，故数列⎩⎪⎨⎪⎧⎭⎪⎬⎪⎫a n n ＋1是以a 12＝3为首项，1为公差的等差数列，故a nn ＋1＝3＋1·(n －1)＝n ＋2，故a n ＝(n ＋1)(n ＋2)． 答案：(n ＋1)(n ＋2)2.(创新题)已知数列{a n }满足a n ＋1＝⎩⎪⎨⎪⎧a n 2(a n 为偶数)，a n －2n (a n 为奇数).若a 3＝1，则a 1的所有可能取值为________．解析：当a 2为奇数时，a 3＝a 2－4＝1，a 2＝5； 当a 2为偶数时，a 3＝12a 2＝1，a 2＝2；当a 1为奇数时，a 2＝a 1－2＝5，a 1＝7 或a 2＝a 1－2＝2，a 1＝4(舍去)； 当a 1为偶数时，a 2＝12a 1＝5，a 1＝10或a 2＝12a 1＝2，a 1＝4.综上，a 1的可能取值为4,7,10. 答案：4,7,103．(2013·南通一模)在数列{a n }中，a 1＝1，a 2＝0，对任意正整数n ，m (n >m )满足a 2n －a 2m ＝a n －m a n ＋m ，则a 119＝________.解析：法一：采用特殊值法求出a 3，a 4，a 5，a 6分别为－1,0,1,0，由不完全归纳法得出a n 的周期为4，所以a 119＝a 29×4＋3＝－1.法二：令m ＝2，得a 2n －a 22＝a n －2·a n ＋2，即a 2n ＝a n －2·a n ＋2，所以奇数项成等比数列，偶数项均为0.再令m ＝1，得a 2n －a 21＝a n －1·a n ＋1，当n 为奇数时，a 2n ＝a 21；当n 为偶数时，a n －1·a n ＋1＝－1，故a 1＝－a 3＝a 5＝－a 7＝…，因此a n 的周期为4，所以a 119＝a 29×4＋3＝－1.答案：－14．(2013·扬州期末)若数列{a n }满足a 1为大于1的常数，a n ＋1－1＝a n (a n －1)(n ∈N *)，且1a 1＋1a 2＋…＋1a 2 012＝2，则a 2 013－4a 1的最小值为________． 解析：因为a 1>1，易知对所有的n ∈N *，a n >1，对a n ＋1－1＝a n (a n －1)两边取倒数得1a n ＋1－1＝1a n (a n －1)＝1a n －1－1a n，所以1a n ＝1a n －1－1a n ＋1－1，所以1a 1＋1a 2＋…＋1a 2 012＝1a 1－1－1a 2 013－1＝2.整理得a 2 013＝2－a 13－2a 1(由a 2 013>1得1<a 1<32)，所以a 2 013－4a 1＝2(3－2a 1)＋12(3－2a 1)－112≥22(3－2a 1)·12(3－2a 1)－112＝－72，当且仅当a 1＝54时取等号．故a 2 013－4a 1的最小值为－72. 答案：－72第二节等差数列及其前n 项和对应学生用书P691．等差数列的有关概念(1)定义：如果一个数列从第2项起，每一项与它的前一项的差都等于同一个常数，那么这个数列就叫做等差数列．符号表示为a n ＋1－a n ＝d (n ∈N *，d 为常数)．(2)等差中项：数列a ，A ，b 成等差数列的充要条件是A ＝a ＋b2，其中A 叫做a ，b 的等差中项．2．等差数列的有关公式 (1)通项公式：a n ＝a 1＋(n －1)d . (2)前n 项和公式：S n ＝na 1＋n (n －1)2d ＝(a 1＋a n )n2.1．要注意概念中的“从第2项起”．如果一个数列不是从第2项起，而是从第3项或第4项起，每一项与它前一项的差是同一个常数，那么此数列不是等差数列．2．注意区分等差数列定义中同一个常数与常数的区别． [试一试]1．在等差数列{a n }中，已知a 4＋a 8＝16，则该数列前11项和S 11＝________. 解析：∵a 4＋a 8＝16， ∴a 6＝8，∴S 11＝11a 6＝88. 答案：882．(2013·重庆高考)已知{a n }是等差数列，a 1＝1，公差d ≠0，S n 为其前n 项和，若a 1，a 2，a 5成等比数列，则S 8＝________.解析：因为{a n }为等差数列，且a 1，a 2，a 5成等比数列，所以a 1(a 1＋4d )＝(a 1＋d )2，解得d ＝2a 1＝2，所以S 8＝64.答案：641．等差数列的四种判断方法(1)定义法：a n ＋1－a n ＝d (d 是常数)⇔{a n }是等差数列． (2)等差中项法：2a n ＋1＝a n ＋a n ＋2(n ∈N *)⇔{a n }是等差数列． (3)通项公式：a n ＝pn ＋q (p ，q 为常数)⇔{a n }是等差数列． (4)前n 项和公式：S n ＝An 2＋Bn (A 、B 为常数)⇔{a n }是等差数列． 2．活用等差数列的常用性质(1)通项公式的推广：a n ＝a m ＋(n －m )d ，(n ，m ∈N *)．(2)若{a n }为等差数列，且k ＋l ＝m ＋n ，(k ，l ，m ，n ∈N *)，则a k ＋a l ＝a m ＋a n .(3)若{a n }是等差数列，公差为d ，则a k ，a k ＋m ，a k ＋2m ，…(k ，m ∈N *)是公差为md 的等差数列．(4)数列S m ，S 2m －S m ，S 3m －S 2m ，…也是等差数列. 3．用方程思想和化归思想在解有关等差数列的问题时可以考虑化归为a 1和d 等基本量，通过建立方程(组)获得解． [练一练]1．(2014·盐城摸底)已知等差数列{a n }满足a 3＋a 7＝10，则该数列的前9项和S 9＝________.解析：由题知，S 9＝9(a 1＋a 9)2＝9(a 3＋a 7)2＝45.答案：452．(2014·南京、盐城一模)在等差数列{a n }中，若a 3＋a 5＋a 7＝9，则其前9项和S 9的值为________．解析：由题知a 3＋a 5＋a 7＝3a 5＝9，则a 5＝3，所以S 9＝9a 5＝27. 答案：27对应学生用书P701.(2013·n n m －12，S m ＝0，S m ＋1＝3，则m ＝________.解析：根据已知条件，得到a m 和a m ＋1，再根据等差数列的定义得到公差d ，最后建立关于a 1和m 的方程组求解．由S m －1＝－2，S m ＝0，S m ＋1＝3，得a m ＝S m －S m －1＝2，a m ＋1＝S m ＋1－S m ＝3， 所以等差数列的公差为d ＝a m ＋1－a m ＝3－2＝1，由⎩⎪⎨⎪⎧a m ＝a 1＋(m －1)d ＝2，S m ＝a 1m ＋12m (m －1)d ＝0，得⎩⎪⎨⎪⎧a 1＋m －1＝2，a 1m ＋12m (m －1)＝0，解得⎩⎪⎨⎪⎧a 1＝－2，m ＝5. 答案：52．(2014·扬州调研)在等差数列{a n }中，若a 1＋a 2＝4，a 9＋a 10＝36，则S 10＝________. 解析：法一：由于a 1＋a 2＋a 9＋a 10＝2(a 1＋a 10)＝40， 故a 1＋a 10＝20，从而S 10＝10(a 1＋a 10)2＝100.法二：由题意得⎩⎪⎨⎪⎧ 2a 1＋d ＝4，2a 1＋17d ＝36，解得⎩⎪⎨⎪⎧a 1＝1，d ＝2，从而S 10＝10a 1＋10×9d2＝100.答案：1003．设{a n }是公差不为零的等差数列，S n 为其前n 项和，且满足a 22＋a 23＝a 24＋a 25，S 7＝7.(1)求数列{a n }的通项公式及前n 项和S n ；(2)试求所有的正整数m ，使得a m a m ＋1a m ＋2为数列{a n }中的项．解：(1)由题意可设等差数列{a n }的通项公式 a n ＝a 1＋(n －1)d ，d ≠0.由a 22＋a 23＝a 24＋a 25化简得2a 1＋5d ＝0.①又因为S 7＝7，所以a 1＋3d ＝1.② 由①②可知a 1＝－5，d ＝2.所以数列{a n }的通项公式为a n ＝2n －7， 其前n 项和S n ＝n (a 1＋a n )2＝n 2－6n .(2)因为a m a m ＋1a m ＋2＝(a m ＋2－4)(a m ＋2－2)a m ＋2＝a m ＋2－6＋8a m ＋2为数列{a n }中的项，则8a m ＋2为整数．又由(1)知a m ＋2＝2m －3为奇数，所以a m ＋2＝2m －3＝±1，解得m ＝1或2. 经检验，符合题意的正整数m ＝2.[备课札记] [类题通法]1．等差数列的通项公式及前n 项和公式共涉及五个量a 1，a n ，d ，n ，S n ，知其中三个就能求另外两个，体现了用方程组解决问题的思想．2．数列的通项公式和前n 项和公式在解题中起到变量代换的作用，而a 1和d 是等差数列的两个基本量，用它们表示已知和未知是常用方法．[典例] 已知数列{a n }的前n 项和为S n ，且满足a 1＝12，a n ＝－2S n S n －1(n ≥2且n ∈N *)．(1)求证：数列⎩⎨⎧⎭⎬⎫1S n 是等差数列．(2)求S n 和a n .[解] (1)证明：当n ≥2时，a n ＝S n －S n －1＝－2S n S n －1，① ∴S n (1＋2S n －1)＝S n －1.由上式知若S n －1≠0，则S n ≠0. ∵S 1＝a 1≠0，由递推关系知S n ≠0(n ∈N *)， 由①式得1S n －1S n －1＝2(n ≥2)．∴⎩⎨⎧⎭⎬⎫1S n 是等差数列，其中首项为1S 1＝1a 1＝2，公差为2.(2)∵1S n ＝1S 1＋2(n －1)＝1a 1＋2(n －1)，∴S n ＝12n.当n ≥2时，a n ＝S n －S n －1＝－12n (n －1)，当n ＝1时，a 1＝S 1＝12不适合上式，∴a n＝⎩⎪⎨⎪⎧12，n ＝1，－12n (n －1)，n ≥2.[备课札记]解：(1)∵S n ＝S n －12S n －1＋1，∴1S n ＝2S n －1＋1S n －1＝1S n －1＋2. ∴1S n －1S n －1＝2. ∴⎩⎨⎧⎭⎬⎫1S n 是以12为首项，以2为公差的等差数列．(2)由(1)知1S n ＝12＋(n －1)×2＝2n －32，即S n ＝12n －32.当n ≥2时，a n ＝S n －S n －1＝12n －32－12n －72 ＝－2⎝⎛⎭⎫2n －32⎝⎛⎭⎫2n －72； 当n ＝1时，a 1＝2不适合a n ， 故a n＝⎩⎪⎨⎪⎧2(n ＝1)，－2⎝⎛⎭⎫2n －32⎝⎛⎭⎫2n －72(n ≥2).[类题通法]1．解答题判断等差数列，常用定义法和等差中项法，而通项公式法和前n 项和公式法主要适用于选择题、填空题中的简单判断．2．用定义证明等差数列时，常采用两个式子a n ＋1－a n ＝d 和a n －a n －1＝d ，但它们的意义不同，后者必须加上“n ≥2”，否则n ＝1时，a 0无定义．[针对训练]在数列{a n }中，a 1＝－3，a n ＝2a n －1＋2n ＋3(n ≥2，且n ∈N *)． (1)求a 2，a 3的值；(2)设b n ＝a n ＋32n (n ∈N *)，证明：{b n }是等差数列．解：(1)∵a 1＝－3，a n ＝2a n －1＋2n ＋3(n ≥2，且n ∈N *)， ∴a 2＝2a 1＋22＋3＝1，a 3＝2a 2＋23＋3＝13. (2)证明：对于任意n ∈N *， ∵b n ＋1－b n ＝a n ＋1＋32n ＋1－a n ＋32n＝12n ＋1[(a n ＋1－2a n )－3]＝12n ＋1[(2n ＋1＋3)－3]＝1， ∴数列{b n }是首项为a 1＋32＝－3＋32＝0，公差为1的等差数列．[典例] n 1352＋a 4＋a 6＝99，{a n }的前n 项和为S n ，则使得S n 达到最大时n ＝________.(2)设数列{a n }，{b n }都是等差数列，若a 1＋b 1＝7，a 3＋b 3＝21，则a 5＋b 5＝________. [解析] (1)a 1＋a 3＋a 5＝105⇒a 3＝35，a 2＋a 4＋a 6＝99⇒a 4＝33，则{a n }的公差d ＝33－35＝－2，a 1＝a 3－2d ＝39，S n ＝－n 2＋40n ，因此当S n 取得最大值时，n ＝20.(2)设两等差数列组成的和数列为{c n }，由题意知新数列仍为等差数列且c 1＝7，c 3＝21，则c 5＝2c 3－c 1＝2×21－7＝35.[答案] (1)20 (2)35[备课札记] [类题通法] 1．等差数列的性质(1)项的性质：在等差数列{a n }中，a m －a n ＝(m －n )d ⇔a m －a n m －n ＝d (m ≠n )，其几何意义是点(n ，a n )，(m ，a m )所在直线的斜率等于等差数列的公差．(2)和的性质：在等差数列{a n }中，S n 为其前n 项和，则 ①S 2n ＝n (a 1＋a 2n )＝…＝n (a n ＋a n ＋1)； ②S 2n －1＝(2n －1)a n .2．求等差数列前n 项和S n 最值的两种方法(1)函数法：利用等差数列前n 项和的函数表达式S n ＝an 2＋bn ，通过配方或借助图像求二次函数最值的方法求解．(2)邻项变号法：①a 1>0，d <0时，满足⎩⎪⎨⎪⎧a m ≥0，a m ＋1≤0的项数m 使得S n 取得最大值为S m ；②当a 1<0，d >0时，满足⎩⎪⎨⎪⎧a m ≤0，a m ＋1≥0的项数m 使得S n 取得最小值为S m .[针对训练]1．设数列{a n }是公差d <0的等差数列，S n 为其前n 项和，若S 6＝5a 1＋10d ，则S n 取最大值时，n ＝________.解析：由题意得S 6＝6a 1＋15d ＝5a 1＋10d ， 所以a 6＝0，故当n ＝5或6时，S n 最大． 答案：5或62．(2013·广东高考)在等差数列{a n }中，已知a 3＋a 8＝10，则3a 5＋a 7＝________.解析：因为a 3＋a 8＝10，所以3a 5＋a 7＝2(a 3＋a 8)＝20. 答案：20对应学生用书P71[课堂练通考点]1．(2013·南京、淮安二模)设数列{a n }是公差不为零的等差数列，S n 为其前n 项和．若a 21＋a 22＝a 23＋a 24，S 5＝5，则a 7的值是________．解析：设数列{a n }的公差为d .由a 21＋a 22＝a 23＋a 24得a 21＋(a 1＋d )2＝(a 1＋2d )2＋(a 1＋3d )2，即8a 1d ＋12d 2＝0.因为d ≠0，所以a 1＝－32d .又由S 5＝5a 3＝5得a 3＝1，所以a 1＋2d ＝1，解得a 1＝－3，d ＝2，故－3＋(7－1)×2＝9.答案：92．设等差数列{a n }的前n 项和为S n ，若a 11－a 8＝3，S 11－S 8＝3，则使a n >0的最小正整数n 的值是________．解析：∵a 11－a 8＝3d ＝3，∴d ＝1，∵S 11－S 8＝a 11＋a 10＋a 9＝3a 1＋27d ＝3， ∴a 1＝－8，∴a n ＝－8＋(n －1)>0， 解得n >9，因此使a n >0的最小正整数n 的值是10. 答案：103．已知数列{a n }为等差数列，S n 为其前n 项和，a 7－a 5＝4，a 11＝21，S k ＝9，则k ＝________. 解析：a 7－a 5＝2d ＝4，则d ＝2.a 1＝a 11－10d ＝21－20＝1， S k ＝k ＋k (k －1)2×2＝k 2＝9.又k ∈N *，故k ＝3.答案：34．已知一等差数列的前四项和为124，后四项和为156，各项和为210，则此等差数列的项数是________．解析：设数列{a n }为该等差数列， 依题意得a 1＋a n ＝124＋1564＝70.∵S n ＝210，S n ＝n (a 1＋a n )2，∴210＝70n2，∴n ＝6.答案：65．各项均为正数的数列{a n }满足a 2n ＝4S n －2a n －1(n ∈N *)，其中S n 为{a n }的前n 项和．(1)求a 1，a 2的值； (2)求数列{a n }的通项公式．解：(1)当n ＝1时，a 21＝4S 1－2a 1－1， 即(a 1－1)2＝0，解得a 1＝1.当n ＝2时，a 22＝4S 2－2a 2－1＝4a 1＋2a 2－1＝3＋2a 2， 解得a 2＝3或a 2＝－1(舍去)． (2)a 2n ＝4S n －2a n －1，① a 2n ＋1＝4S n ＋1－2a n ＋1－1.②②－①得：a 2n ＋1－a 2n ＝4a n ＋1－2a n ＋1＋2a n ＝2(a n ＋1＋a n )，即(a n ＋1－a n )(a n ＋1＋a n )＝2(a n ＋1＋a n )． ∵数列{a n }各项均为正数， ∴a n ＋1＋a n >0，a n ＋1－a n ＝2，∴数列{a n }是首项为1，公差为2的等差数列． ∴a n ＝2n －1.[课下提升考能]第Ⅰ卷：夯基保分卷1．(2014·泰州模拟)在等差数列{a n }中，若a 3＋a 9＋a 27＝12，则a 13＝________. 解析：等差数列{a n }中，由a 3＋a 9＋a 27＝12得3a 13＝12，所以a 13＝4. 答案：42．已知等差数列{a n }满足a 2＝3，S n －S n －3＝51(n >3)，S n ＝100，则n 的值为________． 解析：由S n －S n －3＝51得， a n －2＋a n －1＋a n ＝51， 所以a n －1＝17，又a 2＝3，S n ＝n (a 2＋a n －1)2＝100，解得n ＝10. 答案：103．(2014·镇江月考)已知等差数列{a n }中，a 4＋a 6＝10，前5项和S 5＝5，则其公差为________．解析：由a 4＋a 6＝10，得2a 5＝10，所以a 5＝5.由S 5＝5a 3＝5，得a 3＝1，所以d ＝a 5－a 32＝5－12＝2. 答案：24．已知S n 是等差数列{a n }的前n 项和，S 10>0并且S 11＝0，若S n ≤S k 对n ∈N *恒成立，则正整数k 构成的集合为________．解析：在等差数列{a n }中，由S 10>0，S 11＝0得， S 10＝10(a 1＋a 10)2>0⇒a 1＋a 10>0⇒a 5＋a 6>0， S 11＝11(a 1＋a 11)2＝0⇒a 1＋a 11＝2a 6＝0，故可知等差数列{a n }是递减数列且a 6＝0， 所以S 5＝S 6≥S n ，其中n ∈N *， 所以k ＝5或6. 答案：{5,6}5．(2013·南通二模)设等差数列{a n }的公差为正数，若a 1＋a 2＋a 3＝15，a 1a 2a 3＝80，则a 11＋a 12＋a 13＝________.解析：由条件可知，a 2＝5，从而a 1＋a 3＝10，a 1a 3＝16，得a 1＝2，a 3＝8，公差为3，所以a 11＋a 12＋a 13＝2×3＋(10＋11＋12)×3＝105.答案：1056．(2013·常州质检)设s ，t 为正整数，两条直线l 1：t 2s x ＋y －t ＝0与l 2：t 2s x －y ＝0的交点是(x 1，y 1)，对于正整数n (n ≥2)，过点(0，t )和(x n －1,0)的直线与直线l 2的交点记为(x n ，y n )，则x n －y n ＝________(用s ，t ，n 表示)．解析：法一：点(x n ，y n )满足⎩⎪⎨⎪⎧tx ＋x n －1y ＝tx n －1，t 2s x －y ＝0，得到x n ＝2sx n －12s ＋x n －1，y n ＝tx n －12s ＋x n －1，所以x n －y n ＝(2s －t )x n －12s ＋x n －1.点(x 1，y 1)满足⎩⎨⎧t2sx ＋y －t ＝0，t2s x －y ＝0，解得x 1＝s ，y 1＝t 2，所以x 2＝23s ，y 2＝t 3；x 3＝12s ，y 3＝14t ；x 4＝25s ，y 4＝15t ，… 猜想：x n ＝2s n ＋1，y n ＝t n ＋1.所以x n －y n ＝2s n ＋1－tn ＋1＝2s －t n ＋1.法二：由法一知x 1＝s ，y 1＝t2，x n ＝2sx n －12s ＋x n －1，y n ＝tx n －12s ＋x n －1由2sx n ＋x n x n －1＝2sx n －1可化为2s x n －2s x n －1＝1，即⎩⎨⎧⎭⎬⎫2s x n 是以2sx 1＝2为首项，1为公差的等差数列．所以2s x n ＝2＋(n －1)，得x n ＝2s n ＋1，将其代入y n 得y n ＝tn ＋1，故x n －y n ＝2s －t n ＋1.答案：2s －t n ＋17．(2013·南京二模)设S n 是等差数列{a n }的前n 项和，若S 3S 7＝13，则S 6S 7＝________.解析：由S 3＝3a 2，S 7＝7a 4，S 3S 7＝13得9a 2＝7a 4＝7(a 2＋2d )，即a 2＝7d ，所以a 3＝8d ，a 4＝9d ，从而S 6＝3(a 3＋a 4)＝51d ，S 7＝7a 4＝63d ，故结果为1721.答案：17218．(2013·无锡期末)已知数列{a n }的前n 项和S n ＝n 2－7n ，且满足16<a k ＋a k ＋1<22，则正整数k ＝________.解析：由a n ＝⎩⎪⎨⎪⎧S 1， n ＝1S n －S n －1，n ≥2，可得a n ＝2n －8,16<a k ＋a k ＋1<22，即16<(2k －8)＋(2k－6)<22，所以7.5<k <9，又k ∈N *，所以k ＝8.答案：89．(2013·苏锡常镇、连云港、徐州六市调研(二))已知等差数列{a n }的公差d 不为0，且a 3＝a 27，a 2＝a 4＋a 6.(1)求数列{a n }的通项公式；(2)设数列{a n }的前n 项和为S n ，求满足S n －2a n －20>0的所有正整数n 的集合．解：(1)由a 3＝a 27得a 1＋2d ＝(a 1＋6d )2.① 由a 2＝a 4＋a 6得a 1＋d ＝2a 1＋8d ，即a 1＝－7d .②将②代入①得－5d ＝d 2.所以d ＝－5或d ＝0(不符合题意．舍去)． 则a 1＝35.所以a n ＝35＋(n －1)(－5)＝－5n ＋40. (2)S n ＝(35－5n ＋40)n 2＝n (75－5n )2.不等式S n －2a n －20>0，即n (75－5n )2－2(－5n ＋40)－20>0，整理得n 2－19n ＋40<0. 所以19－2012<n <19＋2012.因为n ∈N *，则19－142≤n ≤19＋142， 即52≤n ≤332. 所以所求n 的集合为{3,4，…，16}．10．(2014·南京学情调研)已知数列{a n }的首项a 1＝a ，S n 是数列{a n }的前n 项和，且满足S 2n ＝3n 2a n ＋S 2n －1，a n ≠0，n ≥2，n ∈N *.(1)若数列{a n }是等差数列，求a 的值；(2)确定a 的取值集合M ，使a ∈M 时，数列{a n }是递增数列．解：(1)在S 2n ＝3n 2a n ＋S 2n －1中分别令n ＝2，n ＝3及a 1＝a 得(a ＋a 2)2＝12a 2＋a 2，(a ＋a 2＋a 3)2＝27a 3＋(a ＋a 2)2.因为a n ≠0，所以a 2＝12－2a ，a 3＝3＋2a .因为数列{a n }是等差数列，所以a 1＋a 3＝2a 2， 即2(12－2a )＝a ＋3＋2a ，解得a ＝3.经检验a ＝3时，a n ＝3n ，S n ＝3n (n ＋1)2，S n －1＝3n (n －1)2满足S 2n ＝3n 2a n ＋S 2n －1. 所以a ＝3.(2)由S 2n ＝3n 2a n ＋S 2n －1得S 2n －S 2n －1＝3n 2a n ，即(S n ＋S n －1)(S n －S n －1)＝3n 2a n ， 故(S n ＋S n －1)a n ＝3n 2a n .因为a n ≠0，所以S n ＋S n －1＝3n 2(n ≥2)， ① 所以S n ＋1＋S n ＝3(n ＋1)2② ②－①得a n ＋1＋a n ＝6n ＋3(n ≥2)． ③ 所以a n ＋2＋a n ＋1＝6n ＋9.④④－③得a n ＋2－a n ＝6(n ≥2)，即数列a 2，a 4，a 6，…及数列a 3，a 5，a 7，…都是公差为6的等差数列． 因为a 2＝12－2a ，a 3＝3＋2a ，所以a n＝⎩⎪⎨⎪⎧a ， n ＝13n ＋2a －6，n 为奇数且n ≥3，3n －2a ＋6，n 为偶数，要使数列{a n }是递增数列，须有a 1<a 2，且当n 为大于或等于3的奇数时，a n <a n ＋1，且当n 为偶数时，a n <a n ＋1，即⎩⎪⎨⎪⎧a <12－2a ，3n ＋2a －6<3(n ＋1)－2a ＋6(n 为大于或等于3的奇数)，3n －2a ＋6<3(n ＋1)＋2a －6(n 为偶数)，解得94<a <154.所以集合M ＝⎩⎨⎧⎭⎬⎫a |94<a <154，当a ∈M 时，数列{a n }是递增数列． 第Ⅱ卷：提能增分卷1．(2013·苏锡常镇、连云港、徐州六市调研(二))设S n ，T n 分别是等差数列{a n }，{b n }的前n 项和，已知S n T n ＝2n ＋14n －2，n ∈N *，则a 10b 3＋b 18＋a 11b 6＋b 15＝________.解析：因为{a n }，{b n }是等差数列，故b 3＋b 18＝b 6＋b 15，所以a 10b 3＋b 18＋a 11b 6＋b 15＝a 10＋a 11b 3＋b 18＝a 1＋a 20b 1＋b 20＝S 20T 20＝2×20＋14×20－2＝4178. 答案：41782．(2014·盐城二模)在等差数列{a n }中，a 2＝5，a 6＝21，记数列{1a n}的前n 项和为S n ，若S 2n ＋1－S n ≤m15对n ∈N *恒成立，则正整数m 的最小值为________．解析：由条件得公差d ＝21－54＝4，从而a 1＝1，所以a n ＝4n －3，数列{1a n}的前n 项和为S n ＝1＋15＋…＋14n －3.原不等式可化为14n ＋1＋14n ＋5＋…＋18n ＋1≤m 15，记f (n )＝14n ＋1＋14n ＋5＋…＋18n ＋1.因为f (n ＋1)－f (n )＝18n ＋9－14n ＋1<0，故f (n )为单调递减数列，从而f (n )max＝f (1)＝15＋19＝1445.由条件得m 15≥1445，解得m ≥143，故正整数m 的最小值为5.答案：53．(2014·南通一模)已知数列{a n }中，a 2＝1，前n 项和为S n ，且S n ＝n (a n －a 1)2.(1)求a 1；(2)求证：数列{a n }为等差数列，并写出其通项公式；(3)设lg b n ＝a n ＋13n ，试问是否存在正整数p ，q (其中1<p <q )，使b 1，b p ，b q 成等比数列？若存在，求出所有满足条件的数组(p ，q )；若不存在，请说明理由．解：(1)令n ＝1，则a 1＝S 1＝1(a 1－a 1)2＝0.(2)证明：由S n ＝n (a n －a 1)2，即S n ＝na n2，①得S n ＋1＝(n ＋1)a n ＋12.② ②－①得(n －1)a n ＋1＝na n ，③于是na n ＋2＝(n ＋1)a n ＋1. ④④－③得na n ＋2＋na n ＝2na n ＋1， 即a n ＋2＋a n ＝2a n ＋1， 又a 1＝0，a 2＝1，a 2－a 1＝1，所以数列{a n }是以0为首项，1为公差的等差数列． 所以a n ＝n －1.(3)假设存在正整数数组(p ，q )使b 1，b p ，b q 成等比数列，则lg b 1，lg b p ，lg b q 成等差数列，于是2p 3p ＝13＋q 3q .所以q ＝3q (2p 3p －13)．(*)易知(p ，q )＝(2,3)为方程(*)的一组解． 当p ≥3，且p ∈N *时，2(p ＋1)3p ＋1－2p 3p ＝2－4p 3p ＋1<0，故数列⎩⎨⎧⎭⎬⎫2p 3p (p ≥3)为递减数列， 于是2p 3p －13≤2×333－13<0，所以此时方程(*)无正整数解．综上，存在唯一正整数数组(p ，q )＝(2,3)，使b 1，b p ，b q 成等比数列．4．(2013·南京、淮安二模)已知数列{a n }的各项都为正数，且对任意n ∈N *，a 2n ＋1＝a n a n ＋2＋k (k为常数)．(1)若k ＝(a 2－a 1)2，求证：a 1，a 2，a 3成等差数列； (2)若k ＝0，且a 2，a 4，a 5成等差数列，求a 2a 1的值；(3)已知a 1＝a ，a 2＝b (a ，b 为常数)，是否存在常数λ，使得a n ＋a n ＋2＝λa n ＋1对任意n ∈N *都成立？若存在，求λ的值；若不存在，请说明理由．解：(1)证明：当k ＝(a 2－a 1)2时，在a 2n ＋1＝a n a n ＋2＋k 中，令n ＝1，得a 22＝a 1a 3＋(a 2－a 1)2， 即a 1a 3－2a 1a 2＋a 21＝0.因为a 1>0，所以a 3－2a 2＋a 1＝0， 即a 2－a 1＝a 3－a 2.故a 1，a 2，a 3成等差数列．(2)当k ＝0时，a 2n ＋1＝a n a n ＋2，n ∈N *.因为数列{a n }的各项都为正数，所以数列{a n }是等比数列． 设公比为q (q >0)．因为a 2，a 4，a 5成等差数列，所以a 2＋a 5＝2a 4， 即a 1q ＋a 1q 4＝2a 1q 3.因为a 1>0，q >0，所以q 3－2q 2＋1＝0. 解得q ＝1或q ＝1＋52(负根舍去)．所以a 2a 1＝q ＝1或a 2a 1＝q ＝1＋52.(3)存在常数λ＝a 2＋b 2－k ab ，使a n ＋a n ＋2＝λa n ＋1.证明如下：因为a 2n ＋1＝a n a n ＋2＋k ，所以a 2n ＝a n －1a n ＋1＋k ，n ≥2，n ∈N *.所以a 2n ＋1－a 2n ＝a n a n ＋2－a n－1a n ＋1，即a n a n ＋2＋a 2n ＝a 2n ＋1＋a n －1a n ＋1.(*)由于a n >0，(*)式两边同除以a n a n ＋1 得a n ＋a n ＋2a n ＋1＝a n －1＋a n ＋1a n .所以a n ＋a n ＋2a n ＋1＝a n －1＋a n ＋1a n ＝…＝a 1＋a 3a 2，即当n ∈N *，都有a n ＋a n ＋2＝a 1＋a 3a 2a n ＋1. 因为a 1＝a ，a 2＝b ，a 2n ＋1＝a n a n ＋2＋k ，所以a 3＝b 2－ka . 所以a 1＋a 3a 2＝a ＋b 2－ka b ＝a 2＋b 2－k ab.所以对任意n ∈N *，都有a n ＋a n ＋2＝λa n ＋1，此时λ＝a 2＋b 2－kab.第三节等比数列及其前n 项和对应学生用书P711．等比数列的有关概念 (1)定义：如果一个数列从第2项起，每一项与它的前一项的比等于同一常数(不为零)，那么这个数列就叫做等比数列．这个常数叫做等比数列的公比，通常用字母q 表示，定义的表达式为a n ＋1a n ＝q .(2)等比中项：如果a 、G 、b 成等比数列，那么G 叫做a 与b 的等比中项．即：G 是a 与b 的等比中项⇔a ，G ，b 成等比数列⇒G 2＝ab .2．等比数列的有关公式 (1)通项公式：a n ＝a 1q n －1.(2)前n 项和公式：S n ＝⎩⎪⎨⎪⎧na 1，q ＝1，a 1(1－q n )1－q＝a 1－a n q 1－q ，q ≠1.1．在等比数列中易忽视每项与公比都不为0.2．在运用等比数列的前n 项和公式时，必须对q ＝1与q ≠1分类讨论，防止因忽略q ＝1这一特殊情形导致解题失误．[试一试]1．在1和9之间插入三个正数，使这五个数成等比数列，则插入的三个数的和为________． 解析：设5个正数的公比为q (q >0)，所以q 4＝91＝9，即q ＝3，则中间3个数的和为q＋q 2＋q 3＝3＋3＋33＝3＋4 3.答案：3＋4 32．(2014·徐州摸底)已知各项均为正数的等比数列{a n }的前n 项和为S n ，若a 3＝18，S 3＝26，则{a n }的公比q ＝________.解析：由⎩⎨⎧a 3＝18，a 1＋a 2＋a 3＝26，q >0得18q 2＋18q＝8，即4q 2－9q －9＝0.所以(4q ＋3)(q －3)＝0.因为q >0，所以q ＝3.答案：31．等比数列的三种判定方法(1)定义：a n ＋1a n＝q (q 是不为零的常数，n ∈N *)⇔{a n }是等比数列．(2)通项公式：a n ＝cq n －1(c 、q 均是不为零的常数，n ∈N *)⇔{a n }是等比数列． (3)等比中项法：a 2n ＋1＝a n ·a n ＋2(a n ·a n ＋1·a n ＋2≠0，n ∈N *)⇔{a n }是等比数列． 2．等比数列的常见性质(1)若m ＋n ＝p ＋q ＝2k (m ，n ，p ，q ，k ∈N *)，则a m ·a n ＝a p ·a q ＝a 2k； (2)若数列{a n }、{b n }(项数相同)是等比数列，则{λa n }、⎩⎨⎧⎭⎬⎫1a n 、{a 2n }、{a n ·b n}、⎩⎨⎧⎭⎬⎫a n bn (λ≠0)仍然是等比数列；(3)在等比数列{a n }中，等距离取出若干项也构成一个等比数列，即a n ，a n ＋k ，a n ＋2k ，a n ＋3k ，…为等比数列，公比为q k ；(4)公比不为－1的等比数列{a n }的前n 项和为S n ，则S n ，S 2n －S n ，S 3n －S 2n 仍成等比数列，其公比为q n ，当公比为－1时，S n ，S 2n －S n ，S 3n －S 2n 不一定构成等比数列．3．求解等比数列的基本量常用的思想方法(1)方程的思想：等比数列的通项公式、前n 项和的公式中联系着五个量：a 1，q ，n ，a n ，S n ，已知其中三个量，可以通过解方程(组)求出另外两个量；其中基本量是a 1与q ，在解题中根据已知条件建立关于a 1与q 的方程或者方程组，是解题的关键．(2)分类讨论思想：在应用等比数列前n 项和公式时，必须分类求和，当q ＝1时，S n ＝na 1；当q ≠1时，S n ＝a 1(1－q n )1－q；在判断等比数列单调性时，也必须对a 1与q 分类讨论．[练一练]1．(2010·江苏高考)函数y ＝x 2(x >0)的图像在点(a k ，a 2k )处的切线与x 轴交点的横坐标为a k ＋1，其中k ∈N *.若a 1＝16，则a 1＋a 3＋a 5的值是________．解析：切线斜率k ＝2a k ，切线方程为 y －a 2k ＝2a k (x －a k )， 即y ＝2a k x －a 2k ，令y ＝0，得x ＝a k2＝a k ＋1，所以{a n }是首项a 1＝16，公比q ＝12的等比数列，所以a n ＝(12)n －5，故a 1＋a 3＋a 5＝21.答案：212．已知数列{a n }是公比q ≠±1的等比数列，则在{a n ＋a n ＋1}，{a n ＋1－a n }，⎩⎨⎧⎭⎬⎫a n a n ＋1，{na n }这四个数列中，是等比数列的有________个．答案：3对应学生用书P72考点一等比数列的基本运算1.(2013·n 264解析：由a 6a 2＝q 4＝16，则q 2＝4，所以有a 4＝a 2q 2＝－8.答案：－82．(2014·扬州模拟)已知等比数列{a n }中，公比q >1，且a 1＋a 4＝9，a 2a 3＝8，则a 2 013＋a 2 014a 2 011＋a 2 012＝________.解析：因为{a n }为等比数列，故a 1a 4＝a 2a 3＝8，与a 1＋a 4＝9联立解得⎩⎪⎨⎪⎧a 1＝1，a 4＝8或⎩⎪⎨⎪⎧a 1＝8，a 4＝1.又q >1，故a 1＝1，a 4＝8，从而q ＝2，故a 2 013＋a 2 014a 2 011＋a 2 012＝q 2＝4. 答案：43．设等比数列{a n }的公比q <1，前n 项和为S n ，已知a 3＝2，S 4＝5S 2，求{a n }的通项公式．解：由题设知a 1≠0，S n ＝a 1(1－q n )1－q，所以⎩⎪⎨⎪⎧a 1q 2＝2， ①a 1(1－q 4)1－q ＝5×a 1(1－q 2)1－q . ②由②式得1－q 4＝5(1－q 2)， 即(q －2)(q ＋2)(q －1)(q ＋1)＝0. 因为q <1，所以q ＝－1，或q ＝－2. 当q ＝－1时，代入①式得a 1＝2， 通项公式a n ＝2×(－1)n －1； 当q ＝－2时，代入①式得a 1＝12，通项公式a n ＝12×(－2)n －1.综上，a n ＝⎩⎪⎨⎪⎧2×(－1)n －1，q ＝－1，12×(－2)n －1，q ＝－2.[备课札记] [类题通法]1．对于等比数列的有关计算问题，可类比等差数列问题进行，在解方程组的过程中要注意“相除”消元的方法，同时要注意整体代入(换元)思想方法的应用．2．在涉及等比数列前n 项和公式时要注意对公比q 是否等于1进行判断和讨论．[典例] n n n n (1)设c n ＝a n －1，求证：{c n }是等比数列； (2)求数列{a n }的通项公式． [解] (1)证明：∵a n ＋S n ＝n ， ① ∴a n ＋1＋S n ＋1＝n ＋1.②②－①得a n ＋1－a n ＋a n ＋1＝1， ∴2a n ＋1＝a n ＋1，∴2(a n ＋1－1)＝a n －1， ∴a n ＋1－1a n －1＝12.∵首项c 1＝a 1－1，又a 1＋a 1＝1， ∴a 1＝12，c 1＝－12.又c n ＝a n －1，故{c n }是以－12为首项，12为公比的等比数列．(2)由(1)知c n ＝－12×⎝⎛⎭⎫12n －1＝－⎝⎛⎭⎫12n ∴a n ＝1－⎝⎛⎭⎫12n.[备课札记]。
《三维设计》高考英语
《三维设计》高考英语随着科技的不断发展，英语已经成为全球使用最广泛的语言之一。
在中国，英语也被视为高考必考科目之一。
然而，对于许多学生来说，英语是一门难以掌握的学科。
为了帮助学生更好地掌握英语，我们推出了一款名为《三维设计》的高考英语复习资料。
《三维设计》高考英语复习资料采用了三维设计的方法，将英语知识进行立体式的整合，以更加直观、形象的方式呈现给学生。
该资料涵盖了高考英语的所有知识点，包括词汇、语法、阅读、写作等方面。
通过大量的例题和练习题，帮助学生加深对英语知识的理解和掌握。
《三维设计》高考英语复习资料具有以下特点：1、针对性强：该资料针对高考英语考试大纲进行编写，涵盖了所有的考试知识点，让学生能够在最短的时间内掌握考试重点和难点。
2、实用性强：该资料注重实用性和实战性，通过大量的例题和练习题，帮助学生加深对英语知识的理解和掌握，提高学生的应试能力。
3、立体式设计：该资料采用了立体式设计的方法，将英语知识进行整合，以更加直观、形象的方式呈现给学生，方便学生记忆和理解。
4、注重基础：该资料注重基础知识的巩固和拓展，通过多种形式的学习方法，帮助学生打好英语基础，提高英语水平。
总之，《三维设计》高考英语复习资料是一款非常实用的复习资料，它能够帮助学生更好地掌握高考英语的考试知识点和应试技巧，提高英语水平，为高考取得好成绩打下坚实的基础。
春季高考英语《大纲短语》春季高考英语《大纲短语》是每位参加春季高考的考生必须掌握的内容，也是英语考试中的重要考点之一。
下面我们将从以下几个方面对《大纲短语》进行详细解析。
一、短语分类《大纲短语》按照使用频率和难度，将英语短语分为三个级别：基础短语、进阶短语和高分短语。
其中，基础短语是考生必须掌握的内容，包括常见的动词短语、介词短语、名词短语等；进阶短语是在基础短语的基础上进行扩展，包括一些常用的形容词短语、副词短语等；高分短语则是难度较大，使用频率较低的短语，包括一些特殊的动词短语、介词短语等。
【三维设计】(江苏专用)高考英语大一轮复习 阅读理解 人物故事类课件
02读文样板
一. 理清文章脉络
本文讲述了作者在英国做心理学家时的一段经历。这段经历告诉我们，有时 给别人心理帮助不一定需要语言的开导。 [1] One day, when I was working as a psychologist in England, an adolescent boy showed up in my office. It was David. He kept walking up and down restlessly, his face pale, and his hands shaking slightly. His head teacher had referred him to me. “This boy has lost his family，” he wrote. “He is understandably very sad and refuses to talk to others, and I'm very worried about him. Can you help？” [2] I looked at David and showed him to a chair. How could I help him? There are problems psychology doesn't have the answer to, and which no words can describe. Sometimes the best thing one can do is to listen openly and sympathetically.
多媒体语篇精读课
——阅读理解解题样板（七）
阅读理解解题样板
01介绍说明类 02科技发明类
三维设计高考英语外研大一轮复习教师用书：必修五 含答案
Module 1British and American EnglishⅠ.单词—在语境中默写，在联想中积累[写得准][用得活][积得多] 1.accent n．口音2.structure n. 结构；体系3.present v t. 陈述；提出(观点、计划等)adj. 目前的；到场的；出席的4.attempt n．& v. 尝试；努力bination n．结合→combine v t.结合6.confusing adj.令人困惑的；难懂的→confuse v t.使困惑→confused adj.感到困惑的pare v t.比较→comparison n．比较；比喻8.variety n．种类→variation n．变化→vary v．改变；使变化→various adj.各种各样的9.differ v i.不同，有区别→difference n．不同；区别→different adj.不同的，有区别的用所给词的适当形式填空1.The confused look on her faceshowed she was confused by theconfusing question.(confuse)2. From the appearance, we cannot tell the differences betweenthe twins, though they alwayshave some different ideas. As forthis phenomenon, ideas differamong many people.(differ)3.He made simple remarks onher remarkableachievements.(remark)4.After he was criticised by thecritic,_he realized that criticismwas very important.(criticise)5.The book explains grammarsimply and clearly, because theauthor used simple examplesentences to simplify abstractconcepts.(simple)bine反义词集锦①separate v t.使分离；使分开②divide v. 分开③divorce v. (使)离婚/分离④isolate v. 隔离；孤立⑤split v t. 分离；使分离2.-ify结尾动词清单①simplify 简化②identify 确定；鉴定；识别③electrify 使电气化；使充电④horrify 使恐惧⑤classify 分类；分等⑥modify 修改，修饰；更改Ⅲ.句式—在解读中学懂，在仿写中学通[背原句][记句式][会仿用] 1.Americans use a flashlight, while forthe British, it's a torch.美国人把手电筒称为flashlight，而英国人叫它torch。
三维设计_2015新课标高考总复习_英语_考点检索A-W
• • • • • • • • • •
at ease 自由自在，安逸 atmosphere [`ætməsfɪə(r)] n. 大气；气氛 attach [ə`tætʃ] v. 把…固定，重视 attack [ə`tæk] vt. / n. 攻击，袭击 attempt [ə`tempt] vt. 试图，尝试 attend [ə`tend] v. 看护，照料，服侍；出席，参加 attract [ə`trækt] v. 吸引，引起 attraction [ə`trækʃ(ə)n] n. 吸引，爱慕 average [`ævərɪdʒ] a.平均；普通的 n.平均数 award [wɔːd] n. 奖品，奖励
（经讨价还价后）成交的商品；
B
• be /become aware of 明白，意识到
Байду номын сангаас
behave [bɪ`heɪv] v. 守规矩，行为 • belief [bɪ`liːf] n. 信条，信念 • belong to 属于
• • • •
•
• • bring up提出；教育；养育；呕出；（船等）停下
• • • • • • • • • •
anyhow [`enɪhaʊ] ad. 不管怎样 apologize [ə`pɔlədʒaɪz] vi. 道歉，谢罪 appeal to [ə`piːl] v. 上诉，申诉，吸引力 applaud [ə`plɔːd] v. / n. 鼓掌,赞许,赞赏 apply [ə`plaɪ] v. 申请 appoint [ə`pɔɪnt] v. 任命，委任，安排，确定 （时间，地点） appreciate [ə`priːʃɪeɪt] v. 欣赏； 感激 approach [ə`prəʊtʃ] n. / v. 靠近，接近，建议，要求 appropriate [ə`prəʊprɪət] a. 合适的，恰当的 approve [ə`pruːv] v.赞成,同意,批准,通过
【三维设计】(江苏专用)高考英语大一轮复习精讲 Unit 3 The world online课件 牛津译林版必修7
包装
服从
(8)book v.
预订
(9)boardv.
(10)behave v．
登上(船、飞机等)
行为
(11)mind v.
介意
(12)serious adj.
(13)crowded adj.
严重的
拥挤的
(14)convenient adj.
(15)abroad adv.
(16)including prep.
was always telling lies.
9．He has only been able to trace out the plan, without any details.
outline (概要) of the
10．I'd like to
withdraw (提取) 1,000 dollars from my savings
reference
(refer) books.
5．The truck was found abandoned (abandon) by the side of a windswept rural road late the next day.
6．A true apology is more than just an (acknowledge) of a mistake.
方便的
在国外
包括
[亮点词汇] (1)accommodation n. 住所，住宿,
(2)manage v．
(3)relieve v.
设法
缓解
(4)regard v.
认为
(5)welleducated adj.
(6)if necessary
背高级表达 2015《三维设计》高考英语(江苏专用)大一轮复习—配套写作专题素材 书面表达必背60个好句子
1.How important and necessary for us to keep the balance of nature and protect the environment! 对于我们来说，维持生态平衡，保护赖以生存的环境 是多么重要啊！
2.We are living in a big family .We should care for and help each other and think it a great pleasure to help others when they are in trouble. 我们共同生活在一个大家庭中。我们应该学会 互相关心，互相帮助，把帮助别人摆脱困扰当 成是一种伟大的志趣。
11. Pollution poses a great threat to our existence. 污染对我们的生存造成巨大威胁。
12. We should do our best to achieve our goal in life. 我们应尽全力去达成我们的人生目标。 13. From what I have mentioned above, we can see
15.There are two reasons for the improvement in people„s living conditions. In the first place, we have been carrying out the reform and opening-up policy. Secondly, there has been a rapid expansion of our national economy. Furthermore, the birth rate has been put under control. 人民生活状况的改善原因有两点。首先，我们一直在贯彻 执行改革开放政策。其次，国民经济正在迅速发展，而且出生 率已经得到控制。
三维设计新课标高考总复习英语考点检索
• demand [dɪ`mɑːnd; (US) dɪ`mænd] vt. 要求
• depend on取决于；依赖；依靠
• deserve [dɪ`zɜːv] v.（不用于进行时态）应得，应受 • design [dɪ`zaɪn] n.& vt. n. 设计，策划 图案 • desire [dɪ`zaɪə(r)] vt. & n. 要求；期望 • desperate [`despərət] a.（因绝望而）不惜冒
• earn [ɜːn] vt. 挣得，赚得 • effect [ɪ`fekt] n. 效果；作用 • elect [ɪ`lekt] vt. （投票）选举
• employ [ɪm`plɔɪ] vt. 雇佣 • entertain [entə‘teɪn] vt. 娱乐；招待；容纳 • envy [·envɪ] vt.& n. 忌妒； 羡慕 • equal [`iːkw(ə)l] a.平等的 vt.等于， • equip [ɪ`kwɪp] vt. 提供设备；装备；配备
• date back to追溯到；从…开始有 • deal with处理；涉及；做生意 • debate [dɪ`beɪt] n. & v.讨论，辩论 • debt [det] n. 债务；欠款 • declare [dɪ`kleə(r)] vt. 声明；断言 • defend [dɪ`fend] vt. 防守；保卫 • delight [dɪ`laɪt] n. 快乐；乐事 • deliver [dɪ`lɪvə(r)] vt. 投递（信件，邮
• escape [ɪ`skeɪp] n.& vi. 逃跑；逃脱
• evidence [`evɪdəns] tʃeɪndʒ]n. 交换，交流
三维设计高考英语外研大一轮复习教师用书：必修五 含答案
Module 1British and American EnglishⅠ.单词—在语境中默写，在联想中积累[写得准][用得活][积得多] 1.accent n．口音2.structure n. 结构；体系3.present v t. 陈述；提出(观点、计划等)adj. 目前的；到场的；出席的4.attempt n．& v. 尝试；努力bination n．结合→combine v t.结合6.confusing adj.令人困惑的；难懂的→confuse v t.使困惑→confused adj.感到困惑的pare v t.比较→comparison n．比较；比喻8.variety n．种类→variation n．变化→vary v．改变；使变化→various adj.各种各样的9.differ v i.不同，有区别→difference n．不同；区别→different adj.不同的，有区别的用所给词的适当形式填空1.The confused look on her faceshowed she was confused by theconfusing question.(confuse)2. From the appearance, we cannot tell the differences betweenthe twins, though they alwayshave some different ideas. As forthis phenomenon, ideas differamong many people.(differ)3.He made simple remarks onher remarkableachievements.(remark)4.After he was criticised by thecritic,_he realized that criticismwas very important.(criticise)5.The book explains grammarsimply and clearly, because theauthor used simple examplesentences to simplify abstractconcepts.(simple)bine反义词集锦①separate v t.使分离；使分开②divide v. 分开③divorce v. (使)离婚/分离④isolate v. 隔离；孤立⑤split v t. 分离；使分离2.-ify结尾动词清单①simplify 简化②identify 确定；鉴定；识别③electrify 使电气化；使充电④horrify 使恐惧⑤classify 分类；分等⑥modify 修改，修饰；更改Ⅲ.句式—在解读中学懂，在仿写中学通[背原句][记句式][会仿用] 1.Americans use a flashlight, while forthe British, it's a torch.美国人把手电筒称为flashlight，而英国人叫它torch。
【学增分技巧】2015《三维设计》高考英语(江苏专用)大一轮复习—配套写作专题课件：4教你如何衔接与过渡
同时 at the same time, in the meantime, meanwhile 过去 at that time, before by that time, earlier, formerly, in the past 现在 at present, presently, right now, nowadays, up to now
表时间关系的关联词
at first, at the beginning of, in the 开始 beginning, to begin with, first(ly), first of all after a while, afterwards, and then, 然后 later, after that, next, shortly after, the next moment at last, at length, finally, eventually, 最终 in the end
表转折的关联词
(al)though, however, on the contrary, but, in spite of, nevertheless, yet, otherwise, despite, while, instead Eg. I covered my ears with hands, but it was no use.
其他 from now on , immediately, right away, right now, shortly, soon, until, when, as soon as, since then, for the time, the moment/minute, hardly...when, no sooner...than, sooner or later, gradually, before long, suddenly Eg. Then I will be happy to be your guide. The next moment the car hit him while he was crossing the road.
【三维设计】2015届高考英语一轮复习 Module 1 My First Day at Senio
课时跟踪检测(一) My First Day at Senior HighⅠ.单项填空1．Operate the electrical appliance following the ________ attached in the box, or you’ll be at risk.A．instructions B．explanationsC．introductions D．descriptions2．(2014·烟台调研)Marty’s disability is “one in a million〞．________， there are not many people in the world like him.A．In other words B．All in allC．For one thing D．On the other hand3．(2014·青岛一模)We were ________ by the ________ news that an eightyearold child went to college.A．amazed; amazed B．amazed; amazingC．amazing; amazing D．amazing; amazed4．What ________ pity that you must leave now! It is such ________ great fun to have dinner with you.A．a; a B．a; theC．/; a D．a; /5．(2014·济南教学质量调研)Nine in ten parents said there were significant differences in their approach to educating their children compared with ________ of their parents.A．those B．oneC．both D．that6．Despite such a big difference in ________ towards what one eats, there is no doubt that people in the West regard the Chinese food as something special.A．point B．ideaC．attitude D．sight7．(2014·潍坊三县联考)—The girl looks so beautiful. Is she a model or a film star? —________.She’s a nurse.A．You bet B．Forget about itC．Far from it D．Whatever you say8．If Joe’s wife doesn’t go to the party，________.A．neither does he B．neither will heC．so does he D．so will he9．(2014·临沂一中月考)I didn’t make his failure public to avoid seeing his ________ look.A．embarrassing B．pleasedC．embarrassed D．pleasing10．(2014·大庆模拟)—How was Robert’s cooking?—Oh, pretty good. I was quite ________.A．admired B．interestedC．impressed D．inspired11．(2014·沈阳市铁路实验中学模拟)—If they ________ about this tomorrow, what shall we do?—We have to advise them not to worry.A．are still worrying B．will worryC．have worried D．would worry12．The town offers entertainments of a kind, but they are ________ what you’ll find in the city.A．something like B．more likeC．anything like D．nothing like13．(2014·哈师大附中模拟)Many reporters hurried to ________ the war as soon as the war broke out.A．cover B．interviewC．discover D．write14．(2014·陕西五校联考)The present I have been looking forward to ________ a big surprise.A．being turning out B．is turned outC．turns out D．turning out15．(2014·陕西宝鸡质检)—What does the model plane look like?—The wings of the plane are ________ of its body.A．more than the length twiceB．twice more than the lengthC．more than twice the lengthD．more twice than the lengthⅡ.完形填空(2014·开封高中模考)People always say that if God closed a door in front of you, there must be a window opened for you.Yes, people always say that, but __1__ is the window for me?I am not disappointed, just a little __2__ or as my roommate said, “Why are you angry with yourself once you __3__ to do anything？〞 I have no idea.I just can’t __4__ me if I can’t understand some easy things.Currently, I just feel __5__ with myself because I couldn’t finish the assignment on the lab, and what was __6__ worse that actually I did not understand what the __7__ was! Fiona asked me why I didn’t __8__ for help.The problem is that if I have to ask some silly questions, I will feel __9__.Before I came here I was confident.But I have totally __10__ it.Sometimes I can’t even find the right words; sometimes I am too shy to __11__ others.I have to admit that I am __12__.The wisest people can always take advantage of others’ __13__.But I was so stupidas to close myself and __14__ a lot of time soaking inside the workbook, which made me more confused.Why can’t I __15__ the sentence? I am __16__ to get help the next time I have any questions.I can __17__ I am not smart, but I shouldn’t hide it.I must __18__ it as soon as possible.__19__ I will have to waste much more time on it — just like tonight.Difficulties always go with me.Go __20__！ I can make it.1.A.which B．whatC．where D．why2．A.worried B．amusedC．excited D．proud3．A.want B．decideC．fail D．like4．A.believe B．likeC．delight D．forgive5．A.unsatisfied B．contentC．pleased D．disappointed6．A.very B．evenC．quite D．fairly7．b B．assignmentC．problem D．question8．A.look B．callC．search D．ask9．A.shamed B．upsetC．comfortable D．sorry10．A.continued B．lostC．changed D．trusted11．A.disturb B．assistC．greet D．puzzle12．A.smart B．cleverC．wrong D．stupid13．A.pressure B．wisdomC．influence D．confidence14．A.take B．costC．kill D．waste15．A.ignore B．guideC．follow D．discover16．A.determined B．willingC．anxious D．ready17．A.declare B．announceC．explain D．admit18．A.answer B．solveC．react D．review19．A.However B．ThusC．Otherwise D．Hopefully20．A.ahead B．onC．down D．upⅢ.阅读理解(2014·山师大附中高考模拟)Ling Ling is worried as Sept. 7, the last day for her to enter the university, approaches. “My dad doe sn’t want to pay for my university education as he believes the investment in higher education can’t be regained，〞the 19yearold told the reporter.Ling was born in a small village in northern Sichuan. Upon completion of their primary school education, her parents started a small business. Five years ago, they moved to Chengdu, the provincial capital of Sichuan, with Ling and her younger brother. They have purchased an apartment and a small shop there. “My family isn’t short of money. My father doesn’t want me to pursue university education because he thinks it not worthwhile going to university，〞 Ling said.Her father estimates it will cost a total of 80,000 yuan in tuition fees and living expenses to finish a fouryear university education. If his daught er works for four years instead, she can earn at least 80,000 yuan in four years. With that money, she can start a small business. “In my neighborhood, at least 10 university graduates cannot find jobs and stay at home，〞 her father told the reporter. He said as a primary school graduate, he earned more than many university graduates with his small business. He said his view that attending a university is not worthwhile is not groundless, pointing that nearly 7 million students graduated from institutions of higher learning nationwide this year and their employment situation is not so good.To persuade her father to support her higher education financially, Ling has done many things but in vain. After her story spread online, a survey was made on the website to seek netizens’ views. More than 10,000 netizens replied and 71 percent supported her father’s view. About 50 percent thought society was a good university and about 20 percent thought a person could improve himself or herself anywhere. Despite strong opposition from survey respondents, 55 readers of them said higher education could broaden one’s horizons. They want to help Ling financially since she cannot borrow money from her relatives who share her father’s view. 1．Why her father doesn’t support Ling’s higher education?A．Because her family is short of money.B．Because she has to enter a local university.C．Because he thinks it not worthwhile going to university.D．Because Ling’s father is a businessman.2．What is the main idea of the passage?A．The worth of university causes heated discussion.B．University will cost much money in tuition fees and living expenses. C．University students’ employment situation is bad.D．Society is a good university.3．According to the view of Ling’s father, their family could at least benefit ________ yuan from her work instead of university in four years.A．80,000 B．160,000 C．10,000 D．18,0004．According to this passage, we know that ________.A．some people consider money more importantB．some people consider education more importantC．the majority of people support Ling Ling’s fatherD．the majority of people support Ling Ling5．Which of the following can be inferred from the last paragraph of the passage?A．Ling is likely to go to university.B．Ling will start a small business.C．Ling will give in.D．Ling doesn’t know how to deal with the situation.答案Ⅰ.1.选 A 句意：遵照附在盒子上的说明书操作这种电器，否如此你会处于危险之中。
三维设计高考英语江苏专用一轮复习素材：高考单词2背佳句记单词
背佳句记单词1. Typical of the grassland dwellers of the continent is the American antelope, or pronghorn. 美洲羚羊，或称叉角羚，是该大陆典型的草原动物。
2. Of the millions who saw Haley’s comet in 1986, how many people will live long enough to see it return in the twenty-first century. 1986年看见哈雷慧星的千百万人当中，有多少人能够长寿到足以目睹它在二十一世纪的回归呢？3. Anthropologists have discovered that fear, happiness, sadness, and surprise are universally reflected in facial expressions.人类学家们已经发现，恐惧，快乐，悲伤和惊奇都会行之于色，这在全人类是共通的。
4. Because of its irritating effect on humans, the use of phenol as a general antiseptic has been largely discontinued. 由于苯酚对人体带有刺激性作用，它基本上已不再被当作常用的防腐剂了。
5. In group to remain in existence, a profit-making organization must, in the long run, produce something consumers consider useful or desirable.任何盈利组织若要生存，最终都必须生产出消费者可用或需要的产品。
6. The greater the population there is in a locality, the greater the need there is for water, transportation, and disposal of refuse. 一个地方的人口越多，其对水，交通和垃圾处理的需求就会越大。
【三维设计】(江苏专用)高考英语大一轮复习 任务型阅读课件
读文样板
任务型阅读解题样板
2013·江苏高考
[1]The everyday signs of conscientiousness ( 认真尽责 ) — being punctual, careful in doing work, selfdisciplined, and scrupulous (一丝不苟的) in attending to responsibilities — are typical characteristics of the model organizational citizen, the people who keep things running as they should. They follow the rules, help out, and are concerned about the people they work with. It's the conscientious worker who helps newcomers or updates people who return after an absence, who gets to work on time and never abuses sick leaves, who always gets things done on deadline.
多媒体语篇精读课
——任务型阅读解题样板
任务型阅读解题样板
精读语篇
任务型阅读解题样板
2013·江苏高考
Quiet Virtue: The Conscientious [1]The everyday signs of conscientiousness (认真尽责) — being punctual, careful in doing work, self-disciplined, and scrupulous ( 一 丝 不 苟 的 ) in attending to responsibilities — are typical characteristics of the model organizational citizen, the people who keep things running as they should. They follow the rules, help out, and are concerned about the people they work with. It's the conscientious worker who helps newcomers or updates people who return after an absence, who gets to work on time and never abuses sick leaves, who always gets things done on deadline. [2]Conscientiousness is a key to success in any field. In studies of job performance, outstanding effectiveness for almost all jobs, from semiskilled labor to sales and management, depends on conscientiousness.It is particularly important for outstanding performance in jobs at the lower levels of an organization: the secretary whose message taking is perfect, the delivery truck driver who is always on time. [3]Among sales representatives for a large American car manufacturer, those who were most conscientious had the largest volume of sales. Conscientiousness also offers a buffer (缓冲) against the threat of job loss in today's constantly changing market, because employees with this quality are among the most valued. For the sales representatives, their level of conscientiousness mattered almost as much as their sales in determining who stayed on.
【三维设计】(江苏专用)高考英语大一轮复习精讲 Unit 1 The world of our senses课件 牛津译林版整理3
[解读] find 语”结构。
herself
staring为“find＋宾语＋宾语补足
[仿写] 我们发现让学生有足够的睡眠很重要。 We find it important to let students have enough sleep.
2．While having dinner, listen to some enjoyable music rather than watch television. 吃饭的时候听一些好听的音乐而不是看电视。
12．roll up
与……相反 盯着看，睁大眼睛看 把……附在……上 伸出(手) 偿还；回报，还击；报复 有意义；讲得通 制订出；算出；进展顺利 邂逅，不期而遇 企盼 当心，留心，密切注意
卷起来
1．Polly found herself staring up at the face of an old man with a beard. 波莉抬头看去，发现是一位长着络腮胡子的老人。
7．It's advisable not to consult the dictionary the moment you
come across
(遇到) a new word. Try to guess at its
meaning.
8．Any significant social change brings about problems and it is essential to work out (制订出) proper solutions.
宝贵的 有意义的 建立
多亏 需要，要求
发起一项运动
(1)convey v． (2)indifferent adj.
[亮点词汇] 传达
【三维设计】(江苏专用)高考英语大一轮复习 阅读理解 新闻广告类课件
01精读语篇
[4] The advanced EP also heats the room evenly, wall to wall and floor to ceiling. It comfortably covers an area up to 350 square feet. Other heaters heat rooms unevenly with most of the heat concentrated to the center of the room. And they only heat an area a few feet around the heater. With the EP, the temperature will not vary in any part of the room. [5] The EP comes with a 3-year warranty (保修) and a 60-day.No questions asked, satisfaction guarantee. If you are not totally satisfied, return it at our expense and your money will be given back to you. [6] Now we have a special offer for 10 days, during which you can enjoy a half price discount and a free delivery. If you order after that, we reserve the right to either accept or reject order requests at the discounted price. [7] Take action right now!
【三维设计】(江苏专用)高考英语大一轮复习精讲 Unit 3 Amazing people课件 牛津译林版整理2
第三层级——背范文、学润色， 让作文洋气十足
(2013· 山东高考)假如你是新华中学的学生李华，你的美国朋 友 Tom 一周前给你发电子邮件，询问你暑假里的打算，但你因准 备期末考试未能及时回复。请根据以下要点给他回封邮件： 1．未及时回信的原因；2.你假期的打算(如做兼职、旅行、做 志愿者等)。 注意：1.词数：120～150；2.可适当增加细节， 以使行文连贯。
Dear Tom， How are you doing? You asked me in your last letter about my plan for the coming summer vacation.I am sorry that I didn't reply without delay. How I wish I had read and replied to your email (虚拟语气) as soon as I received it.However, I was busy preparing for my final exam, which was vital for me (非限制性定语从句)．As a result, I had no choice but to completely focus on my studies.I believe that if you were me, you would make the same choice (虚 拟语气)．
6．Smith doesn't want to be involved in the organization (organize) of the conference, although he is willing to attend and speak.
【世纪金榜】(江苏专用)2015年高考英语大一轮复习 Unit 1、2 Other countrie
【世纪金榜】〔江苏专用〕2015年高考英语大一轮复习 Unit 1、2 Other countries, other cultures Witnessing time课时提升作业牛津译林版选修9Ⅰ. 单项填空1. (2014·马鞍山模拟)All factors, he is sure to be a promising man in the near future.A. being consideredB. to be consideredC. consideredD. considering2. (2014·无锡模拟)I can honestly say with all of my heart that all of my success isto him because I would be nowhere without his help.A. ownedB. owedC. distributedD. devoted3. (2014·南京模拟)—Where is the young man now?—Unable to find a proper job in his country, hehis fortune in a foreign country.A. is seekingB. had soughtC. was seekingD. would seek4. (2014·常州模拟)—Why does our manager look so happy?—He is relieved to know theof the difficult problem.A. judgmentB. evaluationC. settlementD. preparation5. (原创)Premier Li Keqiangthe importance of developing the relationship between China and Thailand while he was visiting it.A. approvedB. devotedC. realizedD. stressed6. (2014·徐州模拟)—I felt very glad that I had passed the exam very smoothly.—The exam was not so difficult, so youyourself with only a little success.A. shouldn’t contentB. can’t have contentedC. oughtn’t to have contentedD. mustn’t have contented7. (2014·镇江模拟)—Tom, why didn’t you answer the door last night?—Oh, I am very sorry. Imy son’s bedroom with some furniture and it was quite noisy.A. equippedB. had equippedC. was equippingD. had been equipping8. (2014·扬州模拟)I still remember in front of our schoolmore than one hundred and fifty meters high.A. stands a towerB. does a tower standC. a tower standD. a tower does stand9. (2014·台州模拟)In Britain and other European countries, certain buildings,particularly beautiful or historically important, are protected by law.A. considerB. consideringC. consideredD. to consider10. (2014·镇江模拟)It is obvious that the boy is influenced by his parents, especiallyhis daily thoughts and behavior.A. in terms ofB. in honor ofC. in favor ofD. in praise of11. (2014·郑州模拟)The boss is away for a meeting, so his secretary will be sent to the airport to meet the guesthim.A. in honor ofB. in charge ofC. in favor ofD. on behalf of12. (2014·泉州模拟)sales of cars up by 30% last year, the traffic in Beijing has become increasingly terrible.A. AsB. SinceC. ForD. With13. —Did you enjoy yourself at the party?—Yes, I’ve never been toone before.A. a more excitedB. the most excitedC. a more excitingD. the most exciting14. (2014·黄山模拟)—Where did you see the movie Life of Pi directed by Ang Lee?—It was in the cinemaI regularly go.A. whichB. thatC. whereD. what15. (2014·西工大附中适应性考试)—Why did you drop the chance of earning big money?—. You know, I don’t want to get rich by taking risks.A. All is well that ends wellB. One man’s meat is another’s poisonC. Better safe than sorryD. No sweet without sweatⅡ. 阅读理解AFor five days, Edmonton’s DowntownPark is transformed into one huge stage where artists are able to share their talents, and where people are able to celebrate and enjoy themselves. Since its beginning in 1980, the Edmonton Folk Music Festival has been commemorating(纪念)the true feeling of what folk music is all about and that’s the traditional togetherness(团结友爱)that is felt when people gather to share stories and feelings through songs.This year will be the sixth year when volunteer Riedel will be offering up her time to the festival. “People coming off a busy spring and summer have a moment of relaxation, 〞Riedel said. “It’s really easy to relax, and it’s great seeing family and friends have fun together. 〞These families and friends come from all different kinds of musical tastes. People who take pleasure in Blues are there, so are people who love Bluegrass. This festival does its best to develop everyone’s musical interests.With so many years of experience, the festival has become a well-oiled machine, and does whatever it can to make attendees feel as comfortable as possible. There are free water stations throughout the venue(举办地)for people to fill up their travel cups. When people buy food, reusable dishes are given a $2 plate fee, but that is returned when the plate is brought back.The festival has completely sold out of tickets, and in record time. But with big names such as Van Morrison and Jakob Dylan, it’s easy to see how that was going to happen. There is no parking area during the festival, so using the Park & Ride system or Edmonton Transit is highly recommended. A bike lock-up area is provided and will be available Thursday until Sunday one hour before the gates open until 45 minutes after the gates close.The Edmonton Folk Music Festival begins on Wednesday, Aug. 4 with Van Morrison playing the special donation fund(基金)concert, and will finish up on Sunday, Aug. 8.1. The Edmonton Folk Music Festival is held mainly to.A. gather people with different musical tastesB. remind people of the real sense of folk musicC. exhibit the good voices of great talents in folk musicD. collect old stories of folk music2. Which of the following is TRUE according to the passage?A. Riedel has volunteered for the festival for at least 5 years.B. It’s hard for people to appreciate Blues.C. It costs people a little to fill up their cups from water stations.D. People have to pay$2 for a plate of food.3. We can learn from the passage that.A. people can get tickets easily for the festivalB. the Edmonton Folk Music Festival is well organizedC. driving one’s own car to the festival is highly recommendedD. bikes are available at the festival from Wednesday to Sunday4. What would be the best title for the passage?A. Folk Music of BluesB. One Festival for AllC. Festival for Family GatheringD. Edmonton’s DowntownParkBThere were smiling children all the way. Clearly they knew at what time the train passed their homes and they made it their business to stand along the railway, wave to complete strangers and cheer them up as they rushed towards Penang. Often whole families stood outside their homes and waved and smiled as if those on the trains were their favorite relatives. This is the simple village people of Malaysia. I was moved.I had always traveled to Malaysia by plane or car, so this was the first time I was on a train. I did not particularlyrelishthe long train journey and had brought along a dozen magazines to read and reread. I looked about the train. There was not one familiar face. I sighed and satdown to read my Economics.It was not long before the train was across the Causeway and in Malaysia. Johore Bahru was just another city like Singapore, so I was tired of looking at the crowds of people as they hurried past. As we went beyond the city, I watched the straight rows of rubber trees and miles and miles of green. Then the first village came into sight. Immediately I came alive, I decided to wave back.From then on my journey became interesting. I threw my magazines into the waste basket and decided to join in Malaysian life. Then everything came alive. The mountains seemed to speak to me. Even the trees were smiling. I stared at everything as if I was looking at it for the first time.The day passed fast and I even forgot to have my lunch until I felt hungry. I looked at my watch and was surprised that it was 3: 00 pm. Soon the train pulled up at Butterworth. I looked at the people all around me. They all looked beautiful. When my uncle arrived with a smile, I threw my arms around him to give him a warm hug(拥抱). I had never done this before. He seemed surprised and then his weather-beaten face warmed up with a huge smile. We walked arm in arm to his car.I looked forward to the return journey.5. The author expected the train trip to be.A. adventurousB. pleasantC. excitingD. dull6. What did the author remember most deeply of her train trip?A. The friendly country people.B. The mountains along the way.C. The crowds of people in the streets.D. The simple lunch served on the train.7. Which of the following words can best take the place of the underlined word“relish〞in the second paragraph?A. chooseB. enjoyC. prepare forD. carry on8. Where was the writer going?A. Johore Bahru.B. The Causeway.C. Butterworth.D. Singapore.9. What can we learn from the story?A. Comfort in traveling by train.B. Pleasure of living in the country.C. Reading gives people delight.D. Smiles brighten people up.【语篇随练】多练一点技高一筹根据阅读理解B回答如下问题1. This was the first time I was on a train.【仿写】这是我第一次来这儿。
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hyper- 在上，超 hyperfrequency超高频 hypersonic超声的
mal- 恶，不良 maltreat虐待 malfunction机能失常 micro- 微 microscope显微镜 microware微波 mid- 中，中间 midair半空中 midstream中流 mini- 小 minibus小公共汽车 ministate小国
be使…加强 belittle使缩小
befriend友好相待
bi二，双 biweekly双周刊
biligual两种语言的
co-, col-, com-, con-, cor- 共同 cooperation协作; collaboration协作,勾结; combine联合; correlation相互关系 counter-, contra反，对应 counteraction反作用 contrast对比,对照
di-, dif-, dis- 否定，相反 diffident不自信的 dislike不喜欢
en-,em- 使… enlarge扩大 enable使…能 empower使…有权力 e-, ex外，出 external外部的
erupt喷出
extra- 以外，超过 extraordinary格外的 extrasolar太阳系以外的
-crat 支持，参与者 democrat民主人士
bureaucrat官僚
-dom 表性质，状态，行为 freedom自由 wisdom智慧
-ee 表动物的承受者 employee雇员 trainee受训练的人 -eer 从事…的人 pioneer开拓者 volunteer志愿者 -en (v ) 使变成 harden使硬 shorten使短 -en (adj ) 有…质的，似…的 golden金色的 woolen毛的 -er, -or 表人，物 singer歌唱家 survivor幸存者
高效记单词
高效记单词系列之二 ——构词记忆法
由一种词类 转化成一种 由两个或更多的 词合成一个词 ,有 词类 .
的用连字符“----” 连接,有的直接写 在一起.
blackboard bus-driver
blue-eyed maybe dark-blue
bedroom
newspaper whenever myself forget-me-not
She cleans her room every day.
词根(又称词干)是一个词的躯体,是不变化部分,
体现该词的基本词义.
前缀(又称词头)是词根之前的部分,它赋予该
词根新的词义,派生新词.
后缀(又称词尾) 是词根之后的部分,它也
赋予该词根以新的词义,派生新词.
英语单词常见前后缀总结
前缀 prefix
trans- 转换，变换，横过，越过 transmit传送 transatlantic横渡大西洋的 tri- 三 tricar三轮车 triangle三角(形)
un- 否定 unstable不稳定的
unknown未知的
under- 下，内(指衣服)，不足 underground地下的 underskirt衬裙 underpay付资不足
1.hand (n)------hand (v) Let’s go hand in hand. Please hand in your your exercise books after class. 2.clean (ad j)-----clean (v) Her room is clean and tidy.
a- ，an①无，不，非 asymmetry(不对称) anhydrous（无水的） ②含有in, on, at, by, with, to等意义 asleep在熟睡中 ahead向前 ab- 脱离 abnormal不正常的 abaxial离开轴心的 anti- 反抗 antiwar反战的 auto- 自动，自己 autobiography自传 autostable自动稳定
post- 后 postwar战后
pre前，领先 prewar战前的
postnatal诞生后的
prefix前缀
re回，再 return返回
restart重新开始
semi- 半 semicircle半圆 semiconductor半导体 sub-, suc-, suf-, sup-次，亚，在下，低于 substandard低于标准规格的 succeed继承 suffix后缀 supplement增补
uni单一 uniform均匀的
unipolar单极的
后缀 suffix
-able,-ible 可…的，能…的 readable可读的 sensible可觉察的 -age表状态，性质，行为 breakage破损 shortage缺乏 -al (adj ) 具有…性质的，如…的 personal个人的 regional地区的 -al (n ) 表动作，人，事物 proposal提案 professional专业人员 -ance, -ence 表状态，行为，性质 importance重要性 confidence自信
mis错，坏 mistake错误
misspell拼错
multi- 多 multiparty多党的
multilingual多种语言的
non否定 nonsmoker不抽烟的人 nonexistent不存 在的
out超过，过度，外，出，除去 outgo走得比…远 outgrow长得太大 outdoor户外的 outroot除根 over上，过度 overwork工作过度 overbridge天桥
-ancy, -ency 表状态，行为，性质 constancy一惯性 urgency紧迫性 -ant, -ent 表人，物，行为 applicant申请人 correspondent通信者 -arian 表各种年龄，派别，主义，职业的人 humanitarian人道主义的 utilitarian功利 主义者
il-, im-, in-, ir- 否定 illogical不合逻辑的 impossible不可能的 invisible 不可见 irrational不合理的 Inter- 互相 interchange交换
interlock连锁
intra-, intro- 在内，内部 intrapersonal个人内心的 introspect内省