错位相减法求和专题训练 2018.1.20
1.已知数列word/media/image1.gif满足word/media/image2.gif,且word/media/image3.gif.
(1)求word/media/image1.gif的通项公式;
(2)设word/media/image4.gif,求数列word/media/image5.gif的前word/media/image6.gif项和word/media/image7.gif;
(3)设word/media/image8.gif,证明: word/media/image9.gif
2.设正项数列word/media/image1.gif的前word/media/image10.gif项和为word/media/image11.gif,且满足word/media/image12.gif,word/media/image13.gif,word/media/image14.gif.
(1)求数列word/media/image1.gif的通项公式;
(2)若正项等比数列word/media/image5.gif满足word/media/image15.gif,且word/media/image16.gif,数列word/media/image17.gif的前word/media/image10.gif项和为word/media/image18.gif.
①求word/media/image18.gif;
②若对任意word/media/image19.gif,word/media/image14.gif,均有word/media/image20.gif恒成立,求实数word/media/image21.gif的取值范围.
3.已知word/media/image14.gif,设word/media/image11.gif是单调递减的等比数列word/media/image1.gif的前word/media/image10.gif项和,word/media/image22.gif且word/media/image23.gif成等差数列.
(1)求数列word/media/image1.gif的通项公式;
(2)记数列word/media/image24.gif的前word/media/image10.gif项和为word/media/image18.gif,求证:对于任意正整数word/media/image10.gif,word/media/image25.gif.
4.递增的等比数列word/media/image1.gif的前word/media/image10.gif项和为word/media/image11.gif,且word/media/image26.gif,word/media/image27.gif.
(1)求数列word/media/image1.gif的通项公式;
(2)若word/media/image28.gif,数列word/media/image5.gif的前word/media/image10.gif项和为word/media/image18.gif,求word/media/image29.gif成立的正整数word/media/image10.gif的最小值.
5.已知数列word/media/image1.gif及word/media/image30.gif,且word/media/image31.gif,word/media/image32.gif.
(1)求word/media/image33.gif的值;
(2)求数列word/media/image1.gif的通项公式;
(3)求证: word/media/image34.gif.
6.已知数列word/media/image1.gif是以2为首项的等差数列,且word/media/image35.gif成等比数列.
(Ⅰ)求数列word/media/image1.gif的通项公式及前word/media/image10.gif项和word/media/image36.gif;
(Ⅱ)若word/media/image37.gif,求数列word/media/image38.gif的前word/media/image10.gif项之和word/media/image39.gif.
7.在数列word/media/image1.gif中,word/media/image40.gif,前word/media/image10.gif项和word/media/image11.gif满足word/media/image41.gif.
(1)求证:当word/media/image19.gif时,数列word/media/image42.gif为等比数列,并求通项公式word/media/image43.gif;
(2)令word/media/image44.gif,求数列word/media/image5.gif的前word/media/image10.gif项和为word/media/image18.gif.
8.已知等差数列word/media/image1.gif的前n项和word/media/image11.gif,且word/media/image45.gif,数列word/media/image5.gif满足word/media/image46.gifword/media/image47.gifword/media/image48.gif.
(1)求数列word/media/image1.gif,word/media/image5.gif的通项公式;
(2)记word/media/image18.gif为数列word/media/image5.gif的前n项和,word/media/image49.gif,试问word/media/image50.gif是否存在最大值,若存在,求出最大值;若不存在,请说明理由.
9.已知数列word/media/image1.gif的前n项和word/media/image51.gif.
(1)求数列word/media/image1.gif的通项公式word/media/image43.gif;
(2)令word/media/image52.gif,求数列word/media/image1.gif的前n项和word/media/image18.gif.
10.已知单调递增的等比数列word/media/image1.gif满足:word/media/image53.gif, word/media/image54.gif
(1)求数列word/media/image1.gif的通项公式;
(2)若word/media/image55.gif,数列word/media/image5.gif的前word/media/image10.gif项和为word/media/image11.gif, word/media/image56.gif成立的正整数word/media/image10.gif的最小值.
参考答案
1.解析:(1)当word/media/image10.gif为奇数时,word/media/image57.gif,此时数列word/media/image58.gif成等差数列. word/media/image59.gif
当word/media/image10.gif当为偶数时,word/media/image60.gif,此时数列word/media/image61.gif成等比数列 word/media/image62.gif
word/media/image63.gif
(2)word/media/image64.gif
word/media/image65.gif
word/media/image66.gif
word/media/image67.gif
word/media/image68.gif
(3) word/media/image70.gif word/media/image71.gif
word/media/image72.gif word/media/image10.gif为奇
word/media/image73.gif word/media/image10.gif为偶
2.解析:(1) word/media/image75.gif,word/media/image76.gif,∴word/media/image77.gif,
∴word/media/image78.gif且各项为正,∴word/media/image79.gif
又word/media/image80.gif,所以word/media/image81.gif,再由word/media/image82.gif得word/media/image83.gif,所以word/media/image84.gif
∴word/media/image85.gif是首项为1,公差为3的等差数列,∴word/media/image86.gif
(2) word/media/image87.gif∴word/media/image88.gif, word/media/image89.gif
①word/media/image90.gif,②word/media/image91.gif
∴word/media/image92.gifword/media/image93.gif, word/media/image94.gif
word/media/image95.gifword/media/image96.gif恒成立
∴word/media/image97.gifword/media/image98.gif,即word/media/image99.gif恒成立.
设word/media/image100.gif, word/media/image101.gif
当word/media/image102.gif时,word/media/image103.gif;word/media/image104.gif时, word/media/image105.gif
∴word/media/image106.gif,∴word/media/image107.gif.
点睛:本题主要考查了数列的综合应用问题,其中解答中涉及到等差数列的通项公式的求解,数列的乘公比错位相减法求和,数列的恒成立的求解等知识点的综合运用,试题有一定的综合性,属于中档试题,解答中准确运算和合理转化恒成立问题是解答的关键.
3.解:(1)设数列word/media/image1.gif的公比word/media/image108.gif,由word/media/image109.gif,
得word/media/image110.gif,
即word/media/image111.gif,∴word/media/image112.gif.word/media/image1.gif是单调递减数列,∴word/media/image113.gif, ∴word/media/image114.gif
(2)由(1)知word/media/image115.gif,
所以word/media/image116.gif,①
word/media/image117.gif,②
②-①得:word/media/image118.gif,
word/media/image119.gif,
由word/media/image120.gif,得word/media/image121.gif,
故word/media/image122.gif
又word/media/image123.gif,因此对于任意正整数word/media/image10.gif, word/media/image25.gif
点睛:本题主要考查了数列的综合应用和不等式关系证明问题,其中解答涉及到等比数列的基本量的运算,数列的乘公比错位相减法求和,以及放缩法证明不等式,突出考查了方程思想和错位相减法求和及放缩法的应用,试题综合性强,属于难题.
4.解析:(1)设等比数列word/media/image1.gif的公比为word/media/image108.gif
由已知,word/media/image124.gif.则word/media/image125.gif,则word/media/image126.gif
两式相除得word/media/image127.gif,
∵数列word/media/image1.gif为递增数列,∴word/media/image62.gif,则word/media/image128.gif,所以word/media/image129.gif.
(2)word/media/image130.gif,
word/media/image131.gif
设word/media/image132.gif,①
word/media/image133.gif,②
①word/media/image134.gif②得:
word/media/image135.gif,
word/media/image136.gif,
word/media/image29.gif, 即word/media/image137.gif,
word/media/image138.gif,∴正整数word/media/image10.gif的最小值是word/media/image139.gif.
点睛:本题主要考查了等比数列的求和公式及通项公式的应用,错位相减求和方法的应用,及指数不等式的求解.
5.解析:(1)由已知word/media/image140.gif,所以word/media/image141.gif.
word/media/image142.gif,所以word/media/image143.gif.
word/media/image144.gif,所以word/media/image145.gif.
(2)令word/media/image146.gif,则word/media/image147.gif,①
word/media/image148.gif,②
两式相减,得
word/media/image149.gifword/media/image150.gif,
所以word/media/image151.gif,即word/media/image152.gif,
又word/media/image141.gif也满足上式,
所以数列word/media/image1.gif的通项公式为word/media/image153.gif.
(3)word/media/image154.gif,
所以word/media/image155.gif,③
word/media/image156.gif,④
①-②得
word/media/image157.gif,
所以word/media/image158.gif. 又word/media/image32.gif,∴word/media/image159.gif,故word/media/image160.gif.
又word/media/image161.gif,
所以word/media/image162.gif是递增数列,故word/media/image163.gif.
所以word/media/image34.gif.
【点睛】本题考查数列的前3项及通项公式的求法,考查不等式的证明,解题时要认真审题,注意错位相减法的合理运用.
6.解析:(Ⅰ) 设数列word/media/image1.gif的公差为word/media/image164.gif,由条件可得word/media/image165.gif,即word/media/image166.gif,
解得word/media/image167.gif或word/media/image168.gif(舍去),
则数列word/media/image1.gif的通项公式为word/media/image169.gif,
word/media/image170.gif.
(Ⅱ)由(Ⅰ)得word/media/image171.gif,
则word/media/image172.gif,①
word/media/image173.gif,②
将①-②得word/media/image174.gif
word/media/image175.gif,
则word/media/image176.gif.
【易错点晴】本题主要考等差数列的通项公式、等比数列的求和公式、以及“错位相减法”求数列的和,属于中档题. “错位相减法”求数列的和是重点也是难点,利用“错位相减法”求数列的和应注意以下几点:①掌握运用“错位相减法”求数列的和的条件(一个等差数列与一个等比数列的积);②相减时注意最后一项的符号;③求和时注意项数别出错;④最后结果一定不能忘记等式两边同时除以word/media/image177.gif.
7.解析:(1)word/media/image178.gif
当word/media/image19.gif时,word/media/image179.gif得word/media/image180.gif,word/media/image181.gif
word/media/image182.gif得 word/media/image183.gif word/media/image184.gifword/media/image185.gif
(2)当word/media/image186.gif时, word/media/image187.gif 当word/media/image19.gif时,word/media/image188.gif
当word/media/image186.gif时, word/media/image189.gif
当word/media/image19.gif时,word/media/image190.gif
令word/media/image191.gif
word/media/image192.gif
word/media/image193.gifword/media/image194.gif word/media/image195.gif word/media/image196.gif
word/media/image197.gif 经检验word/media/image186.gif时,word/media/image198.gif也适合上式.
word/media/image199.gifword/media/image200.gif .
点睛:数列问题是高考中的重要问题,主要考查等差等比数列的通项公式和前word/media/image10.gif项和,主要利用解方程得思想处理通项公式问题,利用分组求和、裂项相消、错位相减法等方法求数列的和.在利用错位相减求和时,要注意提高运算的准确性,防止运算错误.
8.解析:(1)设等差数列word/media/image1.gif的首项为word/media/image201.gif,公差为word/media/image164.gif,
则word/media/image202.gif
由题意得word/media/image203.gif,∴数列word/media/image204.gif是等比数列,且首项和公比都是word/media/image205.gif,word/media/image206.gif.
(2)由(1)得word/media/image207.gif,word/media/image208.gif,
两式相减得:word/media/image209.gif,word/media/image210.gif;
word/media/image211.gif;
word/media/image212.gif
当word/media/image213.gif时,word/media/image214.gif;当word/media/image215.gif时,word/media/image216.gif;
word/media/image217.gif ∴word/media/image50.gif存在最大值为word/media/image218.gif.
点睛:数列问题是高考中的重要问题,主要考查等差等比数列的通项公式和前word/media/image10.gif项和,主要利用解方程得思想处理通项公式问题,利用分组求和、裂项相消、错位相减法等方法求数列的和.在利用错位相减求和时,要注意提高运算的准确性,防止运算错误.
9.解析:(1)当word/media/image186.gif时,word/media/image219.gif;
当word/media/image19.gif时,word/media/image220.gif,word/media/image221.gif也符合,
∴数列word/media/image1.gif的通项公式为word/media/image222.gif.
(2)word/media/image223.gif,
∴word/media/image224.gif
点睛:本题考查了等差数列的定义,求数列的前n项和问题,属于中档题.解决数列的通项公式问题时,一般要紧扣等差等比的定义,利用方程思想求解,数列求和时,一般根据通项的特点选择合适的求和方法,其中裂项相消和错位相减法考查的比较多,主要是对通项的变形转化处理即可.
10.解析:(1)设等比例列word/media/image225.gif的首项为word/media/image226.gif,公比为q
依题意,有word/media/image227.gif,解之得word/media/image228.gif或word/media/image229.gif,
又数列word/media/image1.gif单调递增, word/media/image230.gif
(2)依题意, word/media/image231.gif
word/media/image232.gif①
word/media/image233.gif②
由①—②得: word/media/image234.gif
word/media/image235.gifword/media/image236.gif ,word/media/image237.gif,即word/media/image238.gif,word/media/image239.gif当word/media/image240.gif时,word/media/image241.gif;当word/media/image242.gif时,word/media/image243.gif,word/media/image193.gif使word/media/image56.gif,成立的正整数word/media/image10.gif的最小值为word/media/image139.gif.
【 方法点睛】本题主要考查等比数列的通项公式与求和公式以及错位相减法求数列的的前word/media/image10.gif项和,属于中档题.一般地,如果数列word/media/image1.gif是等差数列,word/media/image5.gif是等比数列,求数列word/media/image244.gif的前word/media/image10.gif项和时,可采用“错位相减法”求和,一般是和式两边同乘以等比数列word/media/image5.gif的公比,然后作差求解, 在写出“word/media/image11.gif”与“word/media/image245.gif” 的表达式时应特别注意将两式“错项对齐”以便下一步准确写出“word/media/image246.gif”的表达式.
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